To my father Ilya Etingof on his 95th birthday with admiration

Introduction↩︎

The purpose of group theory is to give a mathematical treatment of symmetries. For example, symmetries of a set of \(n\) elements form the symmetric group \(S_n\), and symmetries of a regular \(n\)-gon – the dihedral group \(D_n\). Likewise, Lie group theory serves to give a mathematical treatment of continuous symmetries, i.e., families of symmetries continuously depending on several real parameters.

The theory of Lie groups was founded in the second half of the 19th century by the Norwegian mathematician Sophus Lie, after whom it is named. It was then developed by many mathematicians over the last 150 years, and has numerous applications in mathematics and science, especially physics.

A prototypical example of a Lie group is the group \(SO(3)\) of rotational symmetries of the 2-dimensional sphere; in this case the parameters are the Euler angles \(\phi,\theta,\psi\).

It turns out that unlike ordinary parametrized curves and surfaces, Lie groups are determined by their linear approximation at the identity element. This leads to the notion of the Lie algebra of a Lie group. This notion allows one to reformulate the theory of continuous symmetries in purely algebraic terms, which provides an extremely effective way of studying such symmetries. The goal of these notes is to give a detailed study of Lie groups and Lie algebras and interactions between them, with numerous examples.

These notes are based on a year-long introductory course on Lie groups and Lie algebras given by the author at MIT in 2020-2021 (in particular, they contain no original material). The first half (Sections 1-26) corresponds to the first semester and follows rather closely the excellent book “An introduction to Lie groups and Lie algebras" by A. Kirillov Jr. ([1]), but also discusses some additional topics. Namely, after a brief review of geometry and topology of manifolds, it covers the basic theory of Lie groups and Lie algebras, including the three fundamental theorems of Lie theory (except the proof of the third theorem, which is given in the second half). Then it proceeds to nilpotent and solvable Lie algebras, theorems of Lie and Engel, representations of \(\mathfrak{sl}_2\), enveloping algebras and the Poincaré-Birkhoff Witt theorem, free Lie algebras, the Baker-Campbell-Hausdorff formula, and concludes with a detailed study and classification of complex semisimple Lie algebras, their representations, and the Weyl character formula.

The second half (starting with Section 27) covers representation theory of \(GL_n\) and other classical groups, minuscule representations, spin representations and spin groups, representation theory of compact Lie groups (again following [1]) and, more generally, compact topological groups, including existence of the Haar measure and the Peter-Weyl theorem. Then it discusses applications to quantum mechanics (a fairly complete treatment of the hydrogen atom) and proceeds to real forms of semisimple Lie algebras and groups, discussing the classification of such forms in terms of Vogan diagrams, maximal tori and maximal compact subgroups, the polar and Cartan decompositions, and classification of connected compact Lie groups and complex reductive groups. Then we discuss topology of Lie groups and homogeneous spaces (in particular, their cohomology rings), cohomology of Lie algebras, prove the third fundamental theorem of Lie theory and Ado’s theorem on the existence of a faithful representation for a finite dimensional Lie algebra, and conclude with the study of Borel and parabolic subgroups, the flag manifold of a complex semisimple group and the Iwasawa decomposition for real groups.

Some other sources covering the same material are [2]–[5].

Each section roughly corresponds to one 80-minute lecture. Part I consists of 26 sections, which corresponds to a 1-semester course. Part II consists of 25 sections, to allow for a review of Part I. Also, a lot of material is contained in exercises, which are often provided with detailed hints. These exercises were assigned as homework problems.¹

Finally, we note that Lie theory is an inherently synthetic subject. While the main technical tools ultimately boil down to various parts of algebra (notably linear algebra and the theory of noncommutative rings and modules, and, at more advanced stages, algebraic geometry), Lie theory also relies in important ways on analysis, differential equations, differential geometry and topology. Thus, while we try to recall basic notions from these subjects along the way, the reader will need some degree of dexterity with them, which increases as we dig deeper into the material.

Acknowledgments. I’d like to thank David Vogan for inspiring me to write these notes and useful comments, and the students of the MIT courses “Lie groups and Lie algebras, I,II" for feedback. I am especially grateful to Frank Wang and Atticus Wang for careful reading and many corrections to parts I and II, respectively. This work was partially supported by the NSF grant DMS-2001318. The latest version was cleaned up using ChatGPT-5.5.

Lie groups and Lie algebras, I

1 Manifolds↩︎

1.1 Topological spaces and groups↩︎

Recall that the mathematical notion responsible for describing continuity is that of a topological space. Thus, to describe continuous symmetries, we should put this notion together with the notion of a group. This leads to the concept of a topological group.

Recall:

\(\bullet\) A topological space is a set \(X\), certain subsets of which (including \(\emptyset\) and \(X\)) are declared to be open, so that an arbitrary union and finite intersection of open sets is open.

\(\bullet\) The collection of open sets in \(X\) is called the topology of \(X\).

\(\bullet\) A subset \(Z\subset X\) of a topological space \(X\) is closed if its complement is open.

\(\bullet\) If \(X,Y\) are topological spaces then the Cartesian product \(X\times Y\) has a natural product topology in which open sets are (possibly infinite) unions of products \(U\times V\), where \(U\subset X,V\subset Y\) are open.

\(\bullet\) Every subset \(Z\subset X\) of a topological space \(X\) carries a natural induced topology, in which open sets are intersections of open sets in \(X\) with \(Z\).

\(\bullet\) A map \(f: X\to Y\) between topological spaces is continuous if for every open set \(V\subset Y\), the preimage \(f^{-1}(V)\) is open in \(X\).

For example, the open sets of the usual topology of the real line \(\Bbb R\) are (disjoint) unions of open intervals \((a,b)\), where \(-\infty\le a<b\le \infty\).

For example, the group \((\Bbb R,+)\) of real numbers with the operation of addition and the usual topology of \(\Bbb R\) is a topological group, since the functions \((x,y)\mapsto x+y\) and \(x\mapsto -x\) are continuous. Also a subgroup of a topological group is itself a topological group, so another example is rational numbers with addition, \((\Bbb Q,+)\). This last example is not a very good model for continuity, however, and shows that general topological groups are not very well behaved. Thus, we will focus on a special class of topological groups called Lie groups.

Lie groups are distinguished among topological groups by the property that as topological spaces they belong to a very special class called topological manifolds. So we need to start with reviewing this notion.

1.2 Topological manifolds↩︎

Recall:

\(\bullet\) A neighborhood of a point \(x\in X\) in a topological space \(X\) is an open set containing \(x\).

\(\bullet\) A base for a topological space \(X\) is a collection \(\mathcal{B}\) of open sets in \(X\) such that for every neighborhood \(U\) of a point \(x\in X\) there exists a neighborhood \(V\subset U\) of \(x\) which belongs to \(\mathcal{B}\). Equivalently, every open set in \(X\) is a union of members of \(\mathcal{B}\).

For example, open intervals form a base of the usual topology of \(\Bbb R\). Moreover, we may take only intervals whose endpoints have rational coordinates, which gives a countable base for \(\Bbb R\). Also if \(X,Y\) are topological spaces with bases \(\mathcal{B}_X,\mathcal{B}_Y\) then products \(U\times V\), where \(U\in \mathcal{B}_X,V\in \mathcal{B}_Y\), form a base of the product topology of \(X\times Y\). Thus if \(X\) and \(Y\) have countable bases, so does \(X\times Y\); in particular, \(\Bbb R^n\) with its usual (product) topology has a countable base (boxes whose vertices have rational coordinates).

\(\bullet\) \(X\) is Hausdorff if any two distinct points have disjoint neighborhoods.

\(\bullet\) If \(X\) is Hausdorff, we say that a sequence of points \(x_n\in X, n\in \Bbb N\) converges to \(x\in X\) as \(n\to \infty\) (denoted \(x_n\to x\)) if every neighborhood of \(x\) contains almost all terms of this sequence. Then one also says that the limit of \(x_n\) is \(x\) and writes \[\lim_{n\to \infty}x_n=x.\] It is easy to show that the limit is unique when it exists. In a Hausdorff space with a countable base, a closed set is one that is closed under taking limits of sequences.

\(\bullet\) A Hausdorff space \(X\) is compact if every open cover \(\lbrace U_\alpha,\alpha\in A\rbrace\) of \(X\) (i.e., \(U_\alpha\subset X\) for all \(\alpha\in A\) and \(X=\cup_{\alpha\in A}U_\alpha\)) has a finite subcover.

\(\bullet\) A continuous map \(f: X\to Y\) is a homeomorphism if it is a bijection and \(f^{-1}: Y\to X\) is continuous.

The second property is often formulated as the condition that \(X\) is locally homeomorphic to \(\Bbb R^n\).

It is true (although not immediately obvious) that if a nonempty open set in \(\Bbb R^n\) is homeomorphic to one in \(\mathbb{R}^m\) then \(n=m\). Therefore, the number \(n\) is uniquely determined by \(X\) as long as \(X\ne \emptyset\). It is called the dimension of \(X\). (By convention, \(\emptyset\) is a manifold of any integer dimension).

A pair \((U,\phi)\) with the above properties is called a local chart. An atlas of local charts is a collection of charts \((U_\alpha,\phi_\alpha),\alpha\in A\) such that \(\cup_{\alpha\in A}U_\alpha=X;\) i.e., \(\lbrace U_\alpha,\alpha\in A\rbrace\) is an open cover of \(X\). Thus any topological manifold \(X\) admits an atlas labeled by points of \(X\). There are also much smaller atlases. For instance, an open set in \(\Bbb R^n\) has an atlas with just one chart, while the sphere \(S^n\) has an atlas with two charts. Very often \(X\) admits an atlas with finitely many charts. For example, if \(X\) is compact then there is a finite atlas, since every atlas has a finite subatlas. Moreover, there is always a countable atlas, due to the following lemma:

Now let \((U,\phi)\) and \((V,\psi)\) be two charts such that \(V\cap U\ne \emptyset\). Then we have the transition map \[\phi\circ \psi^{-1}: \psi(U\cap V)\to \phi(U\cap V),\] which is a homeomorphism between open subsets in \(\Bbb R^n\). For example, consider the atlas of two charts for the circle \(S^1\) (Example 1(3)), one missing the point \((-1,0)\) and the other missing the point \((1,0)\). Then \(\phi(\theta)=\tan({\theta\over 2})\) and \(\psi(\theta)=\cot({\theta\over 2})\), \(\phi(U\cap V)=\psi(U\cap V)=\Bbb R\setminus 0\), and \((\phi\circ \psi^{-1})(x)=\frac{1}{x}\).

1.3 \(C^k\), real analytic and complex analytic manifolds↩︎

The notion of topological manifold is too general for us, since continuous functions on which it is based in general do not admit a linear approximation. To develop the theory of Lie groups, we need more regularity. So we make the following definition.

It is clear that compatibility is an equivalence relation.

Note that the Cartesian product \(X\times Y\) of manifolds \(X,Y\) is naturally a manifold (of the same regularity type) of dimension \(\dim X+\dim Y\).

1.4 Regular functions↩︎

Now let \(P\in X\) and \((U,\phi)\) be a local chart such that \(P\in U\) and \(\phi(P)=0\). Such a chart is called a coordinate chart around \(P\). In particular, we have local coordinates
\(x_1,...,x_n: U\to \Bbb R\) (or \(U\to \Bbb C\) for complex manifolds), which are just the components of \(\phi\), i.e., \(\phi(Q)=(x_1(Q),...,x_n(Q))\). Note that \(x_i(P)=0\), and \(x_i(Q)\) determine \(Q\) if \(Q\in U\).

In other words, \(f\) is regular if it is expressed as a regular function in local coordinates near every point of \(V\). Clearly, this is independent of the choice of coordinates.

The space (in fact, algebra) of regular functions on \(V\) will be denoted by \(O(V)\).

It is clear that this is an equivalence relation.

The algebra of germs of regular functions at \(P\) is denoted by \(O_P\). Thus we have \(O_P=\underrightarrow{\lim}\;O(U)\), where the direct limit is taken over neighborhoods of \(P\).

1.5 Tangent spaces↩︎

From now on we will only consider smooth, real analytic and complex analytic manifolds. By a derivation at \(P\) we will mean a linear map \(D: O_P\to \Bbb R\) in the smooth and real analytic case and \(D: O_P\to \Bbb C\) in the complex analytic case, satisfying the Leibniz rule \[\label{leib} D(fg)=D(f)g(P)+f(P)D(g).\tag{1}\] Note that for any such \(D\) we have \(D(1)=0\).

Let \(T_PX\) be the space of all such derivations. Thus \(T_PX\) is a real vector space for smooth and real analytic manifolds and a complex vector space for complex manifolds.

Observe that every tangent vector \(v\in T_PX\) defines a derivation \(\partial_v: O(U)\to \Bbb R,\Bbb C\) for every neighborhood \(U\) of \(P\), satisfying 1 . The number \(\partial_vf\) is called the derivative of \(f\) along \(v\). For usual curves and surfaces in \(\Bbb R^3\) these coincide with the familiar notions from calculus.

1.6 Regular maps↩︎

It is easy to see that the composition of regular maps is regular, and that a homeomorphism \(F\) such that \(F,F^{-1}\) are both regular is the same thing as a diffeomorphism (=isomorphism).

Let \(F: X\to Y\) be a regular map and \(P\in X\). Then we can define the differential of \(F\) at \(P\), \(d_PF\), which is a linear map \(T_PX\to T_{F(P)}Y\). Namely, for \(f\in O_{F(P)}\) and \(v\in T_PX\), the vector \(d_PF\cdot v\) is defined by the formula \[(d_PF\cdot v)(f):=v(f\circ F).\] The differential of \(F\) is also denoted by \(F_*\); namely, for \(v\in T_PX\) one writes \(dF_P\cdot v=F_*v\).

Moreover, if \(G: Y\to Z\) is another regular map, then we have the usual chain rule, \[d(G\circ F)_P=dG_{F(P)}\circ dF_P.\] In particular, if \(\gamma: (a,b)\to X\) is a regular parametrized curve then for \(t\in (a,b)\) we can define the velocity vector \(\gamma'(t)\in T_{\gamma(t)}X\) by \[\gamma'(t):=d_t\gamma\cdot 1\] (where \(1\in \Bbb R=T_t(a,b)\)).

1.7 Submersions and immersions, submanifolds↩︎

The following proposition is a version of the implicit function theorem for manifolds.

2 Lie groups, I↩︎

2.1 The definition of a Lie group↩︎

Thus, in a Lie group \(G\) for any \(g\in G\) the left and right translation maps \(L_g,R_g: G\to G\), \(L_g(x):=gx,R_g(x):=xg\), are diffeomorphisms.

Note that any complex Lie group of dimension \(n\) can be regarded as a real Lie group of dimension \(2n\). Also the Cartesian product of real (complex) Lie groups is a real (complex) Lie group.

2.2 Homomorphisms↩︎

We will see later that the last condition is in fact redundant.

2.3 Examples↩︎

2.4 The connected component of \(1\)↩︎

Recall:

\(\bullet\) A topological space \(X\) is path-connected if for any \(P,Q\in X\) there is a continuous map \(x: [0,1]\to X\) such that \(x(0)=P,x(1)=Q\) (such \(x\) is called a path connecting \(P\) to \(Q\)).

\(\bullet\) If \(X\) is any topological space, then for \(P\in X\) we can define its path-connected component to be the set \(X_P\) of \(Q\in X\) for which there is a path connecting \(P\) to \(Q\). Then \(X_P\) is the largest path-connected subset of \(X\) containing \(P\). Clearly, the relation that \(Q\) belongs to \(X_P\) is an equivalence relation, which splits \(X\) into equivalence classes called path-connected components. The set of such components is denoted \(\pi_0(X)\).

\(\bullet\) A topological space \(X\) is connected if the only subsets of \(X\) that are both open and closed are \(\emptyset\) and \(X\). For \(P\in X\), the connected component of \(X\) is the union \(X^P\) of all connected subsets of \(X\) containing \(P\), which is obviously connected itself (so it is the largest connected subset of \(X\) containing \(P\)). A path-connected space \(X\) is always connected but not vice versa (the classic counterexample is the graph of the function \(y=\sin(\frac{1}{x})\) together with the interval \([-1,1]\) of the \(y\)-axis); however, a connected manifold is path-connected (show it!), so for manifolds the notions of connected component and path-connected component coincide.

\(\bullet\) If \(Y\) is a topological space, \(X\) is a set and \(p: Y\to X\) is a surjective map (i.e., \(X=Y/\sim\) is the quotient of \(Y\) by an equivalence relation) then \(X\) acquires a topology called the quotient topology, in which open sets are subsets \(V\subset X\) such that \(p^{-1}(V)\) is open.

Now let \(G\) be a real or complex Lie group, and \(G^\circ\) the connected component of \(1\in G\). Then the connected component of any \(g\in G\) is \(gG^\circ\).

Thus we see that any Lie group is an extension of a discrete countable group by a connected Lie group. This essentially reduces studying Lie groups to studying connected Lie groups. In fact, one can further reduce to simply connected Lie groups, which is done in the next subsections.

3 Lie groups, II↩︎

3.1 A crash course on coverings↩︎

Now we need to review some more topology. Let \(X,Y\) be Hausdorff topological spaces, and
\(p: Y\to X\) a continuous map. Then \(p\) is called a covering if every point \(x\in X\) has a neighborhood \(U\) such that \(p^{-1}(U)\) is a union of disjoint open sets (called sheets of the covering) each of which is mapped homeomorphically onto \(U\) by \(p\):

In other words, there exists a homeomorphism \(h: U\times F\to p^{-1}(U)\) for some discrete space \(F\) with \((p\circ h)(u,f)=u\) for all \(u\in U\), \(f\in F\). I.e., informally speaking, a covering is a map that locally on \(X\) looks like the projection \(X\times F\to X\) for some discrete \(F\).

We will consider only coverings with countable fibers, and just call them coverings. It is clear that a covering of a manifold (\(C^k\), real or complex analytic) is a manifold of the same type, and the covering map is regular.

Two paths \(x_0,x_1: [0,1]\to X\) such that \(x_i(0)=P, x_i(1)=Q\) are said to be homotopic if there is a continuous map \[x: [0,1]\times [0,1]\to X,\] called a homotopy between \(x_0\) and \(x_1\), such that \(x(t,0)=x_0(t)\) and \(x(t,1)=x_1(t)\), \(x(0,s)=P,x(1,s)=Q\). See a movie here:

https://commons.wikimedia.org/wiki/File:Homotopy.gif#/media/File:HomotopySmall.gif

For example, if \(x(t)\) is a path and \(g: [0,1]\to [0,1]\) is a change of parameter with \(g(0)=0\), \(g(1)=1\) then the paths \(x_1(t)=x(t)\) and \(x_2(t)=x(g(t))\) are clearly homotopic.

A path-connected Hausdorff space \(X\) is said to be simply connected if for any \(P,Q\in X\), any paths \(x_0,x_1: [0,1]\to X\) such that \(x_i(0)=P, x_i(1)=Q\) are homotopic.

It is easy to show that any covering has a homotopy lifting property: if \(b\in X\) and \(\widetilde{b}\in p^{-1}(b)\subset Y\) then any path \(\gamma\) starting at \(b\) admits a unique lift to a path \(\widetilde{\gamma}\) starting at \(\widetilde{b}\), i.e., \(p(\widetilde{\gamma})=\gamma\). Moreover, if \(\gamma_1,\gamma_2\) are homotopic paths on \(X\) then \(\widetilde{\gamma}_1,\widetilde{\gamma}_2\) are homotopic on \(Y\) (in particular, have the same endpoint). Thus, if \(Z\) is a simply connected space with a point \(z\) then any continuous map \(f: Z\to X\) with \(f(z)=b\) lifts to a unique continuous map \(\widetilde{f}: Z\to Y\) satisfying \(\widetilde{f}(z)=\widetilde{b}\); i.e., \(p\circ \widetilde{f}=f\). Namely, to compute \(\widetilde{f}(w)\), pick a path \(\beta\) from \(z\) to \(w\), let \(\gamma=f(\beta)\) and consider the path \(\widetilde{\gamma}\). Then the endpoint of \(\widetilde{\gamma}\) is \(\widetilde{f}(w)\), and it does not depend on the choice of \(\beta\).

If \(Z,X\) are manifolds (of any regularity type), \(Z\) is simply connected, and \(f: Z\to X\) is a regular map then the lift \(\widetilde{f}: Z\to Y\) is also regular. Indeed, if we introduce local coordinates on \(Y\) using the homeomorphism between sheets of the covering and their images then \(\widetilde{f}\) and \(f\) will be locally expressed by the same functions.

A covering \(p: Y\to X\) of a path-connected space \(X\) is called universal if \(Y\) is simply connected.

If \(X\) is a sufficiently nice space, e.g., a manifold, its universal covering can be constructed as follows. Fix \(b\in X\) and let \(\widetilde{X}_b\) be the set of homotopy classes of paths on \(X\) starting at \(b\). We have a natural map \(p: \widetilde{X}_b\to X\), \(p(\gamma)=\gamma(1)\). If \(U\subset X\) is a small ball around a point \(x\in X\) then \(U\) is simply connected, so we have a natural identification \(h: U\times F\to p^{-1}(U)\) with \((p\circ h)(u,f)=u\), where \(F=p^{-1}(x)\) is the set of homotopy classes of paths from \(b\) to \(x\); namely, \(h(u,f)\) is the concatenation of \(f\) with any path connecting \(x\) with \(u\) inside \(U\). Here the concatenation \(\gamma_1\circ \gamma_2\) of paths \(\gamma_1,\gamma_2:[0,1]\to X\) with \(\gamma_2(1)=\gamma_1(0)\) is the path \(\gamma=\gamma_1\circ \gamma_2: [0,1]\to X\) such that \(\gamma(t)=\gamma_2(2t)\) for \(t\le 1/2\) and \(\gamma(t)=\gamma_1(2t-1)\) for \(t\ge 1/2\).

The topologies on all such \(p^{-1}(U)\) induced by these identifications glue together into a topology on \(\widetilde{X}_b\), and the map \(p: \widetilde{X}_b\to X\) is then a covering. Moreover, the homotopy lifting property implies that \(\widetilde{X}_b\) is simply connected, so this covering is universal.

It is easy to see that a universal covering \(p: Y\to X\) covers any path-connected covering \(p': Y'\to X\), i.e., there is a covering \(q: Y\to Y'\) such that \(p=p'\circ q\); this is why it is called universal. Therefore a universal covering is unique up to an isomorphism (indeed, if \(Y,Y'\) are universal then we have coverings \(q_1: Y\to Y'\) and \(q_2: Y'\to Y\) and \(q_1\circ q_2=q_2\circ q_1={\rm Id}\)).

Now denote by \(\pi_1(X,x)\) the set of homotopy classes of closed paths on a path-connected space \(X\), starting and ending at \(x\). Then \(\pi_1(X,x)\) is a group under concatenation of paths (concatenation is associative since the paths \(a(bc)\) and \((ab)c\) differ only by parametrization and are hence homotopic). This group is called the fundamental group of \(X\) relative to the point \(x\). It acts on the fiber \(p^{-1}(x)\) for every covering \(p: Y\to X\) (by lifting \(\gamma\in \pi_1(X,x)\) to \(Y\)), which is called the action by deck transformations. This action is transitive iff \(Y\) is path-connected and moreover free iff Y is universal.

Finally, the group \(\pi_1(X,x)\) does not depend on \(x\) up to an isomorphism. More precisely, conjugation by any path from \(x_1\) to \(x_2\) defines an isomorphism \(\pi_1(X,x_1)\to \pi_1(X,x_2)\) (although two non-homotopic paths may define different isomorphisms if \(\pi_1\) is non-abelian).

3.2 Coverings of Lie groups↩︎

Let \(G\) be a connected (real or complex) Lie group and \(\widetilde{G}=\widetilde{G}_1\) be the universal covering of \(G\), consisting of homotopy classes of paths \(x: [0,1]\to G\) with \(x(0)=1\). Then \(\widetilde{G}\) is a group via \((x\cdot y)(t)=x(t)y(t)\), and also a manifold.

This is demonstrated by the famous Dirac belt trick, which illustrates the notion of a spinor; namely, spinors are vectors in \(\Bbb C^2\) acted upon by matrices from \(SU(2)\). Here are some videos of the belt trick:

https://www.youtube.com/watch?v=17Q0tJZcsnY

https://www.youtube.com/watch?v=Vfh21o-JW9Q

3.3 Closed Lie subgroups↩︎

This terminology is justified by the following lemma.

We also have

This theorem is rather nontrivial, and we will not prove it at this time (it will be proved much later in Exercise 237), but we will soon prove a weaker version which suffices for our purposes.

3.4 Generation of connected Lie groups by a neighborhood of the identity↩︎

4 Homogeneous spaces, Lie group actions↩︎

4.1 Homogeneous spaces↩︎

A regular map of manifolds \(p: Y\to X\) is said to be a locally trivial fibration (or fiber bundle) with base \(X\), total space \(Y\) and fiber being a manifold \(F\) if every point \(x\in X\) has a neighborhood \(U\) such that there is a diffeomorphism \(h: U\times F\cong p^{-1}(U)\) with \((p\circ h)(u,f)=u\). In other words, locally \(p\) looks like the projection \(X\times F\to X\) (the trivial fiber bundle with fiber \(F\) over \(X\)), but not necessarily globally so. This generalizes the notion of a covering, in which case \(F\) is \(0\)-dimensional (discrete).

Recall that a sequence of group homomorphisms \(d_i: C^i\to C^{i+1}\) is a complex if for all \(i\), \(d_{i}\circ d_{i-1}\) is the trivial homomorphism \(C^{i-1}\to C^{i+1}\). (One may consider finite complexes, semi-infinite to the left or to the right, or infinite in both directions). In this case \({\rm Im}(d_{i-1})\subset {\rm Ker}(d_i)\) is a subgroup. The \(i\)-th cohomology \(H^i(C^\bullet)\) of the complex \(C^\bullet\) is the quotient \({\rm Ker}(d_i)/{\rm Im}(d_{i-1})\). In general it is just a set but if \(C^i\) are abelian groups, it is also an abelian group. Also recall that a complex \(C^\bullet\) is called exact in the \(i\)-th term if \({\rm Ker}(d_i)={\rm Im}(d_{i-1})\), i.e., if \(H^i(C^\bullet)\) is trivial (consists of one element). A complex exact in all its terms (except possibly first and last, where this condition makes no sense) is called an exact sequence.

4.2 Lie subgroups↩︎

We will call the image of an injective immersion of manifolds an immersed submanifold; it has a manifold structure coming from the source of the immersion.

It is clear that in this case \(H\) is still a Lie group and the inclusion \(H\hookrightarrow G\) is a homomorphism of Lie groups.

We will prove Proposition 18 in Subsection 9.1.

4.3 Actions and representations of Lie groups↩︎

Let \(X\) be a manifold, \(G\) a Lie group, and \(a: G\times X\to X\) a set-theoretical left action of \(G\) on \(X\).

From now on, by an action of \(G\) on \(X\) we will always mean a regular action.

In other words, a representation is a homomorphism of Lie groups \(\pi_V: G\to GL(V)\).

As usual, an isomorphism of representations is an invertible morphism. With these definitions, finite dimensional representations of \(G\) form a category.

Note also that we have the operations of dual and tensor product on representations. Namely, given a representation \(V\) of \(G\), we can define its representation on the dual space \(V^*\) by \[\pi_{V^*}(g)=\pi_V(g^{-1})^*,\] and if \(W\) is another representation of \(G\) then we can define a representation of \(G\) on \(V\otimes W\) (the tensor product of vector spaces) by \[\pi_{V\otimes W}(g)=\pi_V(g)\otimes \pi_W(g).\]

Also if \(V\subset W\) is a subrepresentation (i.e., a subspace invariant under \(G\)) then \(W/V\) is also a representation of \(G\), called the quotient representation.

4.4 Orbits and stabilizers↩︎

As in ordinary group theory, if \(G\) acts on \(X\) and \(x\in X\) then we can define the orbit \(Gx\subset X\) of \(x\) as the set of \(gx\), \(g\in G\), and the stabilizer, or isotropy group \(G_x\subset G\) to be the group of \(g\in G\) such that \(gx=x\).

Proposition 19 will be proved in Subsection 9.1.

4.5 Left translation, right translation, and adjoint action↩︎

Recall that a Lie group \(G\) acts on itself by left translations \(L_g(x)=gx\) and right translations \(R_{g^{-1}}(x)=xg^{-1}\) (note that both are left actions).

Note this is an action by (inner) automorphisms. Also since \({\rm Ad}_g(1)=1\), we have a linear map \(d_1{\rm Ad}_g: \mathfrak{g}\to \mathfrak{g}\), where \(\mathfrak{g}=T_1G\). We will abuse notation and denote this map just by \({\rm Ad}_g\). This defines a representation of \(G\) on \(\mathfrak{g}\) called the adjoint representation.

5 Tensor fields↩︎

5.1 A crash course on vector bundles↩︎

Let \(X\) be a real manifold. A vector bundle on \(X\) is, informally speaking, a (locally trivial) fiber bundle on \(X\) whose fibers are finite dimensional vector spaces. In other words, it is a family of vector spaces parametrized by \(x\in X\) and varying regularly with \(x\). More precisely, we have the following definition.

Let \(\Bbb K=\Bbb R\) or \(\Bbb C\).

In other words, locally on \(X\), \(E\) is isomorphic to \(X\times \Bbb K^n\), but not necessarily globally so.

As for ordinary fiber bundles, \(E\) is called the total space and \(X\) the base of the bundle.

Note that even if \(X\) is a complex manifold and \(\Bbb K=\Bbb C\), \(E\) need not be a complex manifold.

From now on, unless specified otherwise, all complex vector bundles on complex manifolds we consider will be holomorphic.

It follows from the definition that if \(p: E\to X\) is a vector bundle then \(X\) has an open cover \(\lbrace U_\alpha\rbrace\) such that \(E\) trivializes on each \(U_\alpha\), i.e., there is a diffeomorphism \(g_\alpha: U_\alpha\times \Bbb K^n\to p^{-1}(U_\alpha)\) as above. In this case we have clutching functions \[h_{\alpha\beta}: U_\alpha\cap U_\beta\to GL_n(\Bbb K)\] (holomorphic if \(E\) is a holomorphic bundle), defined by the formula \[(g_\alpha^{-1}\circ g_\beta)(x,v)=(x,h_{\alpha\beta}(x)v)\] which satisfy the consistency conditions \[h_{\alpha\beta}(x)=h_{\beta\alpha}(x)^{-1}\] and \[h_{\alpha\beta}(x)\circ h_{\beta\gamma}(x)=h_{\alpha\gamma}(x)\] for \(x\in U_\alpha\cap U_\beta\cap U_\gamma\). Moreover, the bundle can be reconstructed from this data, starting from the disjoint union \(\sqcup_\alpha U_\alpha\times \Bbb K^n\) and identifying (gluing) points according to \[h_{\alpha\beta}: (x,v)\in U_\beta\times \Bbb K^n\sim (x,h_{\alpha\beta}(x)v)\in U_\alpha\times \Bbb K^n.\] The consistency conditions ensure that the relation \(\sim\) is symmetric and transitive, so it is an equivalence relation, and we define \(E\) to be the space of equivalence classes with the quotient topology. Then \(E\) has a natural structure of a vector bundle on \(X\).

This can also be used for constructing vector bundles. Namely, the above construction defines a \(\Bbb K\)-vector bundle on \(X\) once we are given a cover \(\lbrace U_\alpha\rbrace\) on \(X\) and a collection of clutching functions \[h_{\alpha\beta}: U_\alpha\cap U_\beta\to GL_n(\Bbb K)\] satisfying the consistency conditions.

In particular, we may consider sections of a vector bundle \(p: E\to X\) over an open set \(U\subset X\). These sections form a vector space denoted \(\Gamma(U,E)\).

5.2 Vector fields↩︎

Thus in local coordinates a vector field looks like \[\boldsymbol{v}=\sum_i v_i\frac{\partial}{\partial x_i},\] \(v_i=v_i(\boldsymbol{x})\), and if \(x_i\mapsto x_i'\) is a change of local coordinates then the expression for \(\boldsymbol{v}\) in the new coordinates is \[\boldsymbol{v}=\sum_i v_i'\frac{\partial}{\partial x_i'}\] where \[v_i'=\sum_j \frac{\partial x_i'}{\partial x_j}v_j,\] i.e., the clutching function is the Jacobi matrix of the change of variable. Thus, every vector field \(\boldsymbol{v}\) on \(X\) defines a derivation of the algebra \(O(U)\) for every open set \(U\subset X\) compatible with restriction maps \(O(U)\to O(V)\) for \(V\subset U\);⁴ in particular, a derivation \(O_x\to O_x\) for all \(x\in X\). Conversely, it is easy to see that such a collection of derivations gives rise to a vector field, so this is really the same thing.

A manifold \(X\) is called parallelizable if its tangent bundle is trivial. By Exercise 22, this is equivalent to having a collection of vector fields \(\boldsymbol{v}_1,...,\boldsymbol{v}_n\) which form a basis in every tangent space (such a collection is called a frame). For example, the circle \(S^1\) and hence the torus \(S^1\times S^1\) are parallelizable. On the other hand, the sphere \(S^2\) is not parallelizable, since it does not even have a single nowhere vanishing vector field (the Hairy Ball theorem, or Hedgehog theorem). The same is true for any even-dimensional sphere \(S^{2m}\), \(m\ge 1\).

5.3 Tensor fields, differential forms↩︎

Since vector bundles are basically just smooth families of vector spaces varying over some base manifold \(X\), we can do with them the same things we can do with vector spaces - duals, tensor products, symmetric and exterior powers, etc. E.g., the cotangent bundle \(T^*X\) is dual to the tangent bundle \(TX\).

More generally, we make the following definition.

For example, a tensor field of rank \((1,0)\) is a vector field. Also, a skew-symmetric tensor field of rank \((0,m)\) is called a differential \(m\)-form on \(X\). In other words, a differential \(m\)-form is a section of the vector bundle \(\Lambda^mT^*X\).

For instance, if \(f\in O(X)\) then we have a differential \(1\)-form \(df\) on \(X\), called the differential of \(f\) (indeed, recall that \(d_xf: T_xX\to \Bbb K\)). A general \(1\)-form can therefore be written in local coordinates as \[\omega=\sum_i a_idx_i.\] where \(a_i=a_i(\boldsymbol{x})\). If coordinates are changed as \(x_i\mapsto x_i'\), then in new coordinates \[\omega=\sum_i a_i'dx_i'\] where \[a_i'=\sum_j \frac{\partial x_j}{\partial x_i'}a_j.\] Thus the clutching function is the inverse of the Jacobi matrix of the change of variable. For instance, \[df=\sum_i \frac{\partial f}{\partial x_i}dx_i.\]

More generally, a differential \(m\)-form in local coordinates looks like \[\omega=\sum_{1\le i_1<...<i_m\le n}a_{i_1...i_m}(x)dx_{i_1}\wedge...\wedge dx_{i_m}.\]

5.4 Left and right invariant tensor fields on Lie groups↩︎

Note that if a Lie group \(G\) acts on a manifold \(X\), then it automatically acts on the tangent bundle \(TX\) and thus on vector and, more generally, tensor fields on \(X\). In particular, \(G\) acts on tensor fields on itself by left and right translations; we will denote this action by \(L_g\) and \(R_g\), respectively. We say that a tensor field \(T\) on \(G\) is left invariant if \(L_gT=T\) for all \(g\in G\), and right invariant if \(R_gT=T\) for all \(g\in G\).

6 Classical Lie groups↩︎

6.1 First examples of classical groups↩︎

Roughly speaking, classical groups are groups of matrices arising from linear algebra. More precisely, classical groups are the following subgroups of the general linear group \(GL_n(\Bbb K)\): \(GL_n(\Bbb K)\), \(SL_n(\Bbb K)\) (the special linear group), \(O_n(\Bbb K)\), \(SO_n(\Bbb K)\), \(Sp_{2n}(\Bbb K)\), \(O(p,q)\), \(SO(p,q)\), \(U(p,q)\), \(SU(p,q)\),
\(Sp(2p,2q):=Sp_{2n}(\Bbb C)\cap U(2p,2q)\) for \(p+q=n\) (and also some others we’ll consider later).

Namely,

\(\bullet\) The orthogonal group \(O_n(\Bbb K)\) is the group of matrices preserving the nondegenerate quadratic form in \(n\) variables, \(Q=x_1^2+...+x_n^2\) (or, equivalently, the corresponding bilinear form \(x_1y_1+...+x_ny_n\));

\(\bullet\) The symplectic group \(Sp_{2n}(\Bbb K)\) is the group of matrices preserving a nondegenerate skew-symmetric form in \(2n\) variables;

\(\bullet\) The pseudo-orthogonal group \(O(p,q)\), \(p+q=n\) is the group of real matrices preserving a nondegenerate quadratic form of signature \((p,q)\), \(Q=x_1^2+...+x_p^2-x_{p+1}^2-...-x_n^2\) (or, equivalently, the corresponding bilinear form);

\(\bullet\) The pseudo-unitary group \(U(p,q)\), \(p+q=n\) is the group of complex matrices preserving a nondegenerate Hermitian quadratic form of signature \((p,q)\), \(Q=|x_1|^2+...+|x_p|^2-|x_{p+1}|^2-...-|x_n|^2\) (or, equivalently, the corresponding sesquilinear form);

\(\bullet\) The special pseudo-orthogonal, pseudo-unitary, and orthogonal groups \(SO(p,q)\subset O(p,q)\), \(SU(p,q)\subset U(p,q)\), \(SO_n(\Bbb K)\subset O_n(\Bbb K)\) are the subgroups of matrices of determinant \(1\).

Note that the groups don’t change under switching \(p,q\) and that \((S)O_n(\Bbb R)=(S)O(n,0)\); it is also denoted \((S)O(n)\). Also \((S)U(n,0)\) is denoted by \((S)U(n)\).

Let us show that they are all Lie groups. For this purpose we’ll use the exponential map for matrices. Namely, recall from linear algebra that we have an analytic function \({\rm exp}: {\mathfrak{gl}}_n(\Bbb K)\to GL_n(\Bbb K)\) given by the formula \[\exp(a)=\sum_{n=0}^\infty \frac{a^n}{n!},\] and the matrix-valued analytic function log near \(1\in GL_n(\Bbb K)\), \[\log(A)=-\sum_{n=1}^\infty \frac{(1-A)^n}{n}.\] Namely, this is well defined if the spectral radius of \(1-A\) is \(<1\) (i.e., all eigenvalues are in the open unit disk). These maps have the following properties:

(1) They are mutually inverse.

(2) They are conjugation-invariant.

(3) \(d\exp_0=d\log_1={\rm Id}\).

(4) If \(xy=yx\) then \(\exp(x+y)=\exp(x)\exp(y)\). If \(XY=YX\) then \(\log(XY)=\log(X)+\log(Y)\) (for \(X,Y\) sufficiently close to \(1\)).

(5) For \(x\in {\mathfrak{gl}}_n(\Bbb K)\) the map \(t\mapsto \exp(tx)\) is a homomorphism of Lie groups \(\Bbb K\to \mathop{GL}_n(\Bbb K)\).

(6) \(\det \exp(a)=\exp ({\rm Tr}\;a)\), \(\log (\det A)={\rm Tr}(\log A)\).

Now we can look at classical groups and see what happens to the equations defining them when we apply \(\log\).

1. \(G=SL_n(\Bbb K)\). We already showed that it is a Lie group in Example 11(1) but let us re-do it by a different method. The group \(G\) is defined by the equation \(\det A=1\). So for \(A\) close to \(1\) we have \(\log(\det A)=0\), i.e., \({\rm Tr}\log(A)=0\). So \(\log(A)\in \mathfrak{sl}_n(\Bbb K)=\mathfrak{g}\), the space of matrices with trace \(0\). This defines a local chart near \(1\in G\), showing that \(G\) is a manifold (namely, local charts near other points are obtained by translation), hence a Lie group (multiplication is inherited from the ambient matrix group, hence regular in the induced manifold structure).

2. \(G=O_n(\Bbb K)\). The equation is \(A^T=A^{-1}\), thus \(\log(A)^T=-\log(A)\), so \(\log(A)\in \mathfrak{so}_n(\Bbb K)=\mathfrak{g}\), the space of skew-symmetric matrices.

3. \(G=U(n)\). The equation is \(\overline{A}^T=A^{-1}\), thus \(\overline{\log(A)}^T=-\log(A)\), so \(\log(A)\in \mathfrak{u}_n=\mathfrak{g}\), the space of skew-Hermitian matrices.

We see that the logarithm map identifies the neighborhood of \(1\) in the group \(G\) with a neighborhood of \(0\) in a finite-dimensional vector space. Thus we obtain

6.2 Quaternions↩︎

An important role in the theory of Lie groups is played by the algebra of quaternions, which is the only noncommutative finite dimensional division algebra over \(\Bbb R\), discovered in the 19th century by W. R. Hamilton.

This algebra is associative but not commutative.

Given a quaternion \[\boldsymbol{q}=a+b\boldsymbol{i}+c\boldsymbol{j}+d\boldsymbol{k},\;a,b,c,d\in \Bbb R,\] we define the conjugate quaternion by the formula \[\overline{\boldsymbol{q}}=a-b\boldsymbol{i}-c\boldsymbol{j}-d\boldsymbol{k}.\] Thus \[\boldsymbol{q}\overline{\boldsymbol{q}}=|\boldsymbol{q}|^2=a^2+b^2+c^2+d^2\in \Bbb R,\] where \(|\boldsymbol{q}|\) is the length of \(\boldsymbol{q}\) as a vector in \(\Bbb R^4\). So if \(\boldsymbol{q}\ne 0\) then it is invertible and \[\boldsymbol{q}^{-1}=\frac{\overline{\boldsymbol{q}}}{|\boldsymbol{q}|^2}.\]

Thus \(\Bbb H\) is a division algebra (i.e., a skew-field). One can show that the only finite dimensional associative division algebras over \(\Bbb R\) are \(\Bbb R\), \(\Bbb C\) and \(\Bbb H\). (See Exercise 32).

In particular, we can do linear algebra over \(\Bbb H\) in almost the same way as we do over ordinary fields. Namely, every (left or right) module over \(\Bbb H\) is free and has a basis; such a module is called a (left or right) quaternionic vector space. In particular, any (say, right) quaternionic vector space of dimension \(n\) (i.e., with basis of \(n\) elements) is isomorphic to \(\Bbb H^n\). Moreover, \(\Bbb H\)-linear maps between such spaces are given by left multiplication by quaternionic matrices. Finally, it is easy to see that Gaussian elimination works the same way as over ordinary fields; in particular, every invertible square matrix over \(\Bbb H\) is a product of elementary matrices of the form \(1+(\boldsymbol{q}-1)E_{ii}\) and \(1+\boldsymbol{q} E_{ij}\), \(i\ne j\), where \(\boldsymbol{q}\in \Bbb H\) is nonzero.

Also it is easy to show that \[\overline{\boldsymbol{q}_1\boldsymbol{q}_2}=\overline{\boldsymbol{q}_2}\cdot \overline{\boldsymbol{q}_1},\;|\boldsymbol{q}_1\boldsymbol{q}_2|=|\boldsymbol{q}_1|\cdot |\boldsymbol{q}_2|\] (check this!). So quaternions are similar to complex numbers, except they are non-commutative. Finally, note that \(\Bbb H\) contains a copy of \(\Bbb C\) spanned by \(1,\boldsymbol{i}\); however, this does not make \(\Bbb H\) a \(\Bbb C\)-algebra since \(\boldsymbol{i}\) is not a central element.

This is another way to see that \(SU(2)\cong S^3\) as a manifold (since the set of unit quaternions is manifestly \(S^3\)).

This is the quaternionic analog of the trigonometric form of complex numbers, except the “phase" factor \(\frac{\boldsymbol{q}}{|\boldsymbol{q}|}\) is now not in \(S^1\) but in \(S^3=SU(2)\).

6.3 More classical groups↩︎

Now we can define a new classical group \(GL_n(\Bbb H)\), a real Lie group of dimension \(4n^2\), called the quaternionic general linear group. For example, as we just showed, \(GL_1(\Bbb H)=\Bbb H^\times\cong SU(2)\times \Bbb R_{>0}\).

For \(A\in GL_n(\Bbb H)\), let \(\det A\) be the determinant of \(A\) as a linear operator on \(\Bbb C^{2n}=\Bbb H^n\).

Let \(SL_n(\Bbb H)\subset GL_n(\Bbb H)\) be the subgroup of matrices \(A\) with \(\det A=1\), called the quaternionic special linear group.

Thus \(SL_n(\Bbb H)\) is a real Lie group of dimension \(4n^2-1\).

We can also define groups of quaternionic matrices preserving various sesquilinear forms. Namely, let \(V\cong \Bbb H^n\) be a right quaternionic vector space.

Note that the order of factors is important here!

In (i), the pair \((p,q)\) is called the signature of the quaternionic Hermitian form.

Thus the group of symmetries of a nondegenerate quaternionic Hermitian form of signature \((p,q)\) is \(Sp(2p,2q)=Sp_{2n}(\Bbb C)\cap U(2p,2q)\). It is called the quaternionic pseudo-unitary group.

One also sometimes uses the notation \(U(p,q,\Bbb R)=O(p,q)\), \(U(p,q,\Bbb C)=U(p,q)\), \(U(p,q,\Bbb H)=Sp(2p,2q)\), and \(U(n,0,\Bbb K)=U(n,\Bbb K)\) for \(\Bbb K=\Bbb R,\Bbb C,\Bbb H\).

Thus the group of symmetries of a nondegenerate quaternionic skew-Hermitian form is \(O_{2n}(\Bbb C)\cap U(n,n)\). This group is denoted by \(O^*(2n)\) and called the quaternionic orthogonal group. There is also the subgroup \(SO^*(2n)\subset O^*(2n)\) of matrices of determinant \(1\) (having index \(2\)).

All of these groups are Lie groups, which is shown similarly to Subsection 6.1, using the exponential map.

7 The exponential map of a Lie group↩︎

7.1 The exponential map↩︎

We will now generalize the exponential and logarithm maps from matrix groups to arbitrary Lie groups.

Let \(G\) be a real Lie group, \(\mathfrak{g}=T_1G\).

Thus \(\gamma_x(t)=\exp(tx)\). So we have

The following theorem describes the basic properties of the exponential map. Let \(G\) be a real or complex Lie group.

Thus \(\exp\) has an inverse \(\log: U\to \mathfrak{g}\) defined on a neighborhood \(U\) of \(1\in G\) with \(\log(1)=0\). This map is called the logarithm. For \(GL_n(\Bbb K)\) and its Lie subgroups it coincides with the matrix logarithm. The logarithm map defines a canonical coordinate chart on \(G\) near \(1\), so a choice of a basis of \(\mathfrak{g}\) gives a local coordinate system.

7.2 The commutator↩︎

In general (say, for \(G=GL_n(\Bbb K)\), \(n\ge 2\)), \(\exp(x+y)\ne \exp(x)\exp(y)\). So let us consider the map \[(x,y)\mapsto \mu(x,y)=\log(\exp(x)\exp(y))\] which maps \(U\times U\to \mathfrak{g}\), where \(U\subset \mathfrak{g}\) is a neighborhood of \(0\). This map expresses the product in \(G\) in the coordinate chart coming from the logarithm map. We have \(\mu(x,0)=\mu(0,x)=x\) and \(\mu_*(x,y)=x+y\). So, since \(\mu\) is regular, we have the second Taylor approximation \[\mu(x,y)=x+y+\tfrac{1}{2}\mu_2(x,y)+...\] where \(\mu_2=d^2\mu_{(0,0)}\) is the quadratic part and \(...\) are higher terms. Moreover, \(\mu_2(x,0)=\mu_2(0,y)=0\), hence \(\mu_2\) is a bilinear map \(\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}\). It is easy to see that \(\mu(x,-x)=0\), hence \(\mu_2\) is skew-symmetric.

Thus we have \[\label{prodexp} \exp(x)\exp(y)=\exp(x+y+\tfrac{1}{2}[x,y]+...).\tag{3}\]

For \(x\in \mathfrak{g}\) define the linear map \({\rm ad} x: \mathfrak{g}\to \mathfrak{g}\) by \[{\rm ad}x(y)=[x,y].\]

8 Lie algebras↩︎

8.1 The Jacobi identity↩︎

The matrix commutator \([x,y]=xy-yx\) obviously satisfies the identity \[[[x,y],z]+[[y,z],x]+[[z,x],y]=0\] called the Jacobi identity. Thus it is satisfied for any Lie subgroup of \(GL_n(\Bbb K)\).

8.2 Lie algebras↩︎

Thus we have

We will denote the Lie algebra \(\mathfrak{g}=T_1G\) by \({\rm Lie}G\) or \({\rm Lie}(G)\) and call it the Lie algebra of \(G\). We see that the assignment \(G\mapsto {\rm Lie}G\) is a functor from the category of Lie groups to the category of Lie algebras. Thus we have a map \(\mathrm{Hom}(G,K)\to \mathrm{Hom}({\rm Lie} G,{\rm Lie}K)\), which is injective if \(G\) is connected.

Motivated by Proposition 44(v), a Lie algebra \(\mathfrak{g}\) is said to be commutative or abelian if \([x,y]=0\) for all \(x,y\in \mathfrak{g}\).

8.3 Lie subalgebras and ideals↩︎

A Lie subalgebra of a Lie algebra \(\mathfrak{g}\) is a subspace \(\mathfrak{h}\subset \mathfrak{g}\) closed under the commutator. It is called a Lie ideal if moreover \([\mathfrak{g},\mathfrak{h}]\subset \mathfrak{h}\).

8.4 The Lie algebra of vector fields↩︎

Recall that a vector field on a manifold \(X\) is a compatible family of derivations \(\boldsymbol{v}: O(U)\to O(U)\) for open subsets \(U\subset X\).

Thus, the space \({\rm Vect}(X)\) of vector fields on \(X\) is a Lie algebra under the operation \[\boldsymbol{v},\boldsymbol{w}\mapsto [\boldsymbol{v},\boldsymbol{w}],\] called the Lie bracket of vector fields.⁶

In local coordinates we have \[\boldsymbol{v}=\sum_i v_i\frac{\partial}{\partial x_i},\;\boldsymbol{w}=\sum w_j\frac{\partial}{\partial x_j},\] so \[[\boldsymbol{v},\boldsymbol{w}]=\sum_i\left(\sum_j (v_j\tfrac{\partial w_i}{\partial x_j}- w_j\tfrac{\partial v_i}{\partial x_j})\right)\tfrac{\partial}{\partial x_i}.\] This implies that if vector fields \(\boldsymbol{v},\boldsymbol{w}\) are tangent to a \(k\)-dimensional submanifold \(Y\subset X\) then so is their Lie bracket \([\boldsymbol{v},\boldsymbol{w}]\). Indeed, in local coordinates \(Y\) is given by equations \(x_{k+1}=...=x_n=0\), and in such coordinates a vector field is tangent to \(Y\) iff it does not contain terms with \(\tfrac{\partial}{\partial x_j}\) for \(j>k\).

Now let \(G\) be a Lie group and \(\mathrm{Vect}_L(G),\mathrm{Vect}_R(G)\subset \mathrm{Vect}(G)\) be the subspaces of left and right invariant vector fields.

9 Fundamental theorems of Lie theory↩︎

9.1 Proofs of Theorem 13, Proposition 19, Proposition 18↩︎

Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\) and \(X\) be a manifold with an action \(a: G\times X\to X\). Then for any \(z\in \mathfrak{g}\) we have a vector field \(a_*(z)\) on \(X\) given by \[(a_*(z)f)(x)=\tfrac{d}{dt}|_{t=0}f(\exp(-tz)x),\] where \(t\in \Bbb R\), \(f\in O(U)\) for some open set \(U\subset X\) and \(x\in U\).

This motivates the following definition.

Thus an action of a Lie group \(G\) on \(X\) induces an action of the Lie algebra \(\mathfrak{g}={\rm Lie}G\) on \(X\).

Now let \(x\in X\). Then we have a linear map \(a_{*x}: \mathfrak{g}\to T_xX\) given by \(a_{*x}(z):=a_*(z)(x)\).

Part (i) of Theorem 55 is the promised weaker version of Theorem 13 sufficient for our purposes. Also, part (ii) implies Proposition 19.

9.2 The center of \(G\) and \(\mathfrak{g}\)↩︎

Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\) and \(Z=Z(G)\) the center of \(G\), i.e. the set of \(z\in G\) such that \(zg=gz\) for all \(g\in G\). Also let \(\mathfrak{z}=\mathfrak{z}(\mathfrak{g})\) be the set of \(x\in \mathfrak{g}\) such that \([x,y]=0\) for all \(y\in \mathfrak{g}\); it is called the center of \(\mathfrak{g}\).

It is clear that \(G/Z(G)\) is naturally isomorphic to the image of the adjoint representation \({\rm Ad}: G\to GL(\mathfrak{g})\), which motivates the terminology.

9.3 The statements of the fundamental theorems of Lie theory↩︎

These theorems hold for real as well as complex Lie groups. Thus we have

We will discuss proofs of the fundamental theorems of Lie theory in Subsection 10.2. The third theorem is the hardest one, and we will give its complete proof only in Section 49.

9.4 Complexification of real Lie groups and real forms of complex Lie groups↩︎

Let \(\mathfrak{k}\) be a real Lie algebra. Then \(\mathfrak{k}_{\Bbb C}:=\mathfrak{k}\otimes_{\Bbb R}\Bbb C\) is a complex Lie algebra. We say that \(\mathfrak{g}:=\mathfrak{k}_{\Bbb C}\) is the complexification of \(\mathfrak{k}\), and \(\mathfrak{k}\) is a real form of \(\mathfrak{g}\). Thus a real form of \(\mathfrak{g}\) is a real Lie subalgebra \(\mathfrak{k}\subset \mathfrak{g}\) such that the natural map \(\mathfrak{k}\otimes_{\Bbb R}\Bbb C\to \mathfrak{g}\) is an isomorphism.

In this case we have an antilinear involution \(\sigma: \mathfrak{g}\to \mathfrak{g}\) given by \(\sigma(a+ib)=a-ib\) for \(a,b\in \mathfrak{k}\), and \(\mathfrak{k}:=\mathfrak{g}^\sigma\) is the set of fixed points of \(\sigma\). Conversely, it is easy to see that if \(\sigma\) is an antilinear involution of a complex Lie algebra \(\mathfrak{g}\) (i.e., an automorphism as a real Lie algebra such that \(\sigma^2=1\) and \(\sigma(\lambda a)=\overline{\lambda} \sigma(a)\) for \(a\in \mathfrak{g},\lambda\in \Bbb C\)), then \(\mathfrak{k}:=\mathfrak{g}^\sigma\subset \mathfrak{g}\) is a real form of \(\mathfrak{g}\). Thus real forms of a complex Lie algebra are in natural bijection with its antilinear involutions.

Note that two non-isomorphic real Lie algebras can have isomorphic complexifications; in other words, the same complex Lie algebra can have non-isomorphic real forms. For example, \[\mathfrak{u}(n)_{\Bbb C}\cong \mathfrak{gl}_n(\Bbb R)_{\Bbb C}\cong \mathfrak{gl}_n(\Bbb C)\] while for \(n>1\), \[\mathfrak{u}(n)\ncong \mathfrak{gl}_n(\Bbb R),\] since in the first algebra any element \(x\) with nilpotent \({\rm ad}x\) must be zero, while in the second one it does not have to.

Let us now discuss real forms of complex Lie groups. By analogy with the case of Lie algebras, we make the following definition.

Note that a real Lie group \(K\) may not admit a complexification. For example, Exercise 72 shows that this happens if \(K^\circ\cong \widetilde{SL_2(\Bbb R)}\), the universal cover of \(SL_2(\Bbb R)\). On the other hand, Example 23 shows that \(K\) may admit several (in fact, infinitely many) non-isomorphic complexifications.

For example, both \(U(n)\) and \(GL_n(\Bbb R)\) are real forms of \(GL_n(\Bbb C)\), with \(\sigma(g)=\overline{g}\) and \(\sigma(g)=(\overline{g}^T)^{-1}\) respectively. Note that \(GL_n(\Bbb R)\) is not connected, so a real form of a connected Lie group may be disconnected.

We see that every real form (i.e., antilinear involution) of \(\mathfrak{g}\) defines at most one such form for \(G\). However, it could be none since the involution \(\sigma: \mathfrak{g}\to \mathfrak{g}\) may not lift to \(G\). This is demonstrated by the following example.

However, if \(G\) is a simply connected complex Lie group, then every real form of \(\mathfrak{g}\) necessarily defines one for \(G\). Indeed, in this case by the second fundamental theorem of Lie theory (for real Lie groups), the antilinear involution \(\sigma:\mathfrak{g}\to \mathfrak{g}\) lifts to an antiholomorphic involution \(G\to G\).

10 Proofs of the fundamental theorems of Lie theory↩︎

10.1 Distributions and the Frobenius theorem↩︎

The proofs of the fundamental theorems of Lie theory are based on the notion of an integrable distribution in differential geometry, and the Frobenius theorem about such distributions.

This means that in every tangent space \(T_xX\) we fix a \(k\)-dimensional subspace \(D_x\) which varies regularly with \(x\). In other words, on some neighborhood \(U\subset X\) of every \(x\in X\), \(D\) is spanned by vector fields \(\boldsymbol{v}_1,...,\boldsymbol{v}_k\) linearly independent at every point of \(U\).

By definition, every \(x\in X\) is contained in a \(k\)-dimensional disk \(B_x\subset X\) tangent to \(D\) at all its points.

Now let \(D\) be a \(k\)-dimensional integrable distribution on a manifold \(X\). For \(x,y\in X\), let us say that \(x\sim_D y\) if \(x\) can be connected to \(y\) by a piecewise smooth curve whose tangent vector at every point where it is defined belongs to \(D\). This is clearly an equivalence relation. Denote the equivalence class of \(x\) under this relation by \(S_x\).

Choose a countable cover \(\{U_i\}_{i\in \Bbb N}\) of \(X\) by sufficiently small coordinate balls for \(D\) such that every nonempty intersection \(U_i\cap U_j\) is connected. Then for every \(i\), the set \(U_i\cap S_x\) is a union of pairwise disjoint \(k\)-dimensional disks tangent to \(D\) (in the chosen coordinates these are the sets with fixed values of \(x_{k+1},\dots,x_n\)). Call these disks the plaques of \(S_x\) in \(U_i\).

We endow \(S_x\) with the topology for which all plaques are open and carry their usual topology. Then the inclusion \(S_x\hookrightarrow X\) is continuous (so this topology may be stronger than the induced topology). Then \(S_x\) is Hausdorff since so is \(X\).

The immersed submanifold \(S_x\subset X\) is called the integral submanifold for \(D\) through \(x\).

However, for \(k\ge 2\) a distribution is not always integrable.

10.2 Proofs of the fundamental theorems of Lie theory↩︎

10.2.1 Proof of Theorem 58↩︎

Let \(G\) be a Lie group with Lie algebra \(\mathfrak{g}\). Let \(\mathfrak{h}\subset \mathfrak{g}\) be a Lie subalgebra. We need to show that there is a unique (not necessarily closed) connected Lie subgroup \(H\subset G\) with Lie algebra \(\mathfrak{h}\). The proof of existence of \(H\) is based on the Frobenius theorem.

Define the distribution \(D\) on \(G\) by left-translating \(\mathfrak{h}\subset \mathfrak{g}=T_1G\), i.e., \(D_g=L_g\mathfrak{h}\). So any vector field contained in \(D\) is of the form \[\boldsymbol{v}=\sum f_i\boldsymbol{L}_{a_i},\] where \(a_i\) is a basis of \(\mathfrak{h}\) and \(f_i\) are regular functions. Now if \[\boldsymbol{w}=\sum g_j\boldsymbol{L}_{a_j}\] is another such field then \[[\boldsymbol{v},\boldsymbol{w}]=\sum_{i,j}(f_i\boldsymbol{L}_{a_i}(g_j)\boldsymbol{L}_{a_j}-g_j\boldsymbol{L}_{a_j}(f_i)\boldsymbol{L}_{a_i}+f_ig_j[\boldsymbol{L}_{a_i},\boldsymbol{L}_{a_j}]).\] But \([a_i,a_j]=\sum_k c_{ij}^ka_k\), so \[[\boldsymbol{L}_{a_i},\boldsymbol{L}_{a_j}]=\sum_k c_{ij}^k\boldsymbol{L}_{a_k}.\] Thus if \(\boldsymbol{v},\boldsymbol{w}\) are contained in \(D\) then so is \([\boldsymbol{v},\boldsymbol{w}]\). Hence by the Frobenius theorem, \(D\) is integrable.

Now consider the integral (immersed) submanifold \(H\) of \(D\) going through \(1\in G\). We claim that \(H\) is a Lie subgroup of \(G\) with Lie algebra \(\mathfrak{h}\). Indeed, it suffices to show that \(H\) is a subgroup of \(G\). But this is clear since \(H\) is the collection of elements of \(G\) of the form \[g=\exp(a_1)...\exp(a_m),\] where \(a_i\in \mathfrak{h}\).

Moreover, \(H\) is unique since it has to be generated by the image of the exponential map \(\exp: \mathfrak{h}\to G\).

10.2.2 Proof of Theorem 59↩︎

We need to show that the natural map \(\mathrm{Hom}(G,K)\to \mathrm{Hom}({\rm Lie}G,{\rm Lie}K)\) is a bijection if \(G\) is simply connected.

We know this map is injective so we only need to establish surjectivity. For any morphism \(\psi: {\rm Lie}G\to {\rm Lie}K\), consider the morphism \[\theta=({\rm id},\psi): {\rm Lie}G\to {\rm Lie}(G\times K)={\rm Lie}G\oplus {\rm Lie}K\] The previous proposition implies that there is a connected Lie subgroup \(H\subset G\times K\) whose Lie algebra is \({\rm Im}\theta\). We have projection homomorphisms \(p_1: H\to G\), \(p_2: H\to K\), and \((p_1)_*={\rm id}\), so by Proposition 14(ii) \(p_1\) is a covering. Since \(G\) is simply connected, \(p_1\) is an isomorphism, so we can define \(\phi:=p_2\circ p_1^{-1}: G\to K\), and it is easy to see that \(\psi=\phi_*\).

10.2.3 Proof of Theorem 60↩︎

Finally, let us discuss a proof of Theorem 60, stating that any finite dimensional Lie algebra \(\mathfrak{g}\) over \(\Bbb K=\Bbb R\) or \(\Bbb C\) is the Lie algebra of a Lie group. We will deduce it from the following purely algebraic Ado’s theorem.

Ado’s theorem in fact holds over any ground field, but it is rather nontrivial and we won’t prove it now. A proof can be found, for example, in [6]. But Ado’s theorem immediately implies Theorem 60. Indeed, using Theorem 58, Ado’s theorem implies the following even stronger statement:

This implies

However, it is not true that any Lie group is isomorphic to a Lie subgroup of \(GL_n(\Bbb K)\), see Exercise 72.

One can also prove Theorem 60 directly and then deduce Ado’s theorem as a corollary. We will do this in Sections 49 and 50. We note that Theorem 60 will not be used in proofs of other results until that point.

11 Representations of Lie groups and Lie algebras↩︎

11.1 Representations↩︎

We have previously defined (finite dimensional) representations of Lie groups and (iso)morphisms between them. We can do the same for Lie algebras:

The first and second fundamental theorems of Lie theory imply:

As usual, a subrepresentation of a representation \(V\) is a subspace \(W\subset V\) invariant under the \(G\)-action (resp. \(\mathfrak{g}\)-action). In this case the quotient space \(V/W\) has a natural structure of a representation, called the quotient representation. The notion of direct sum of representations is defined in an obvious way: \[\rho_{V\oplus W}=\rho_V\oplus \rho_W.\] Also we have the notion of dual representation: \[\rho_{V^*}(g)=\rho_V(g^{-1})^*, g\in G;\;\rho_{V^*}(x)=-\rho_V(x)^*, x\in \mathfrak{g},\] and tensor product: \[\rho_{V\otimes W}(g)=\rho_V(g)\otimes \rho_W(g),\;\rho_{V\otimes W}(x)=\rho_V(x)\otimes 1_W+1_V\otimes \rho_W(x).\] Thus we have the notion of symmetric and exterior powers \(S^mV,\wedge^m V\) of a representation \(V\), which can be defined either as quotients or (over a field of characteristic zero) as subrepresentations of \(V^{\otimes m}\). Also for representations \(V,W\), \(\mathrm{Hom}(V,W)\) is a representation via \[g\circ A=\rho_W(g)A\rho_V(g^{-1}),\;x\circ A=\rho_W(x)A-A\rho_V(x),\] so if \(V\) is finite dimensional then \(\mathrm{Hom}(V,W)\cong V^*\otimes W\). Finally, for every representation \(V\) we have the notion of invariants: \[V^G=\lbrace v\in V: gv=v\;\forall g\in G\rbrace,\;V^\mathfrak{g}=\lbrace v\in V: xv=0\;\forall x\in \mathfrak{g}\rbrace.\] Thus \(V^G\subset V^\mathfrak{g}\) and \(V^G=V^\mathfrak{g}\) for connected \(G\) (in general, \(V^G=(V^\mathfrak{g})^{G/G^\circ}\)). Also \(\mathrm{Hom}(V,W)^G\cong \mathrm{Hom}_G(V,W)\) and \(\mathrm{Hom}(V,W)^\mathfrak{g}=\mathrm{Hom}_\mathfrak{g}(V,W)\), the spaces of intertwining operators. Note that in all cases the formula for Lie algebras is determined by the formula for groups by the requirement that these definitions should be consistent with the assignment \(\rho\mapsto \rho_*\).

It is clear that any finite dimensional representation is isomorphic to a direct sum of indecomposable representations (in fact, uniquely so up to order of summands by the Krull-Schmidt theorem). However, not any \(V\) is a direct sum of irreducible representations, e.g. \[\rho: \Bbb C\to GL_2(\Bbb C),\;\rho(x)=\begin{pmatrix} 1& x\\ 0& 1\end{pmatrix}.\]

Some of the main problems of representation theory are:

1) Classify irreducible representations;

2) If \(V\) is a completely reducible representation, find its decomposition into irreducibles.

3) For which \(G\) are all representations completely reducible?

11.2 Schur’s lemma↩︎

Note that the second statement of Schur’s lemma (unlike the first one) does not hold over \(\Bbb R\). For example, consider the rotation group \(SO(2)\) (or any of its finite subgroups of order \(>2\)) acting on \(V=\Bbb R^2\) by rotations. Then \(\mathrm{End}(V)=\Bbb C\ne \Bbb R\). Similarly, if \(V\) is the representation of \(SU(2)\) on \(\Bbb H\) defined by right multiplication by unit quaternions then \(V\) is an irreducible real representation but \(\mathrm{End}(V)=\Bbb H\ne \Bbb R\). For this reason, in representation theory of Lie groups and Lie algebras one usually considers complex representations. Thus from now on all representations we consider will be assumed complex unless specified otherwise.⁹

11.3 Unitary representations↩︎

A finite dimensional representation \(V\) of \(G\) is said to be unitary if it is equipped with a positive definite Hermitian inner product \(B(,)\) invariant under \(G\), i.e., \(B(gv,gw)=B(v,w)\) for \(v, w\in V\), \(g\in G\).

11.4 Representations of \(\mathfrak{sl}_2\)↩︎

The Lie algebra \(\mathfrak{sl}_2=\mathfrak{sl}_2(\Bbb C)\) has basis \[e=\begin{pmatrix} 0& 1\\ 0& 0\end{pmatrix},\; h=\begin{pmatrix} 1& 0\\ 0& -1\end{pmatrix},\; f=\begin{pmatrix} 0& 0\\ 1& 0\end{pmatrix}\] with commutator \[[e,f]=h,\;[h,e]=2e,\;[h,f]=-2f.\] Since 2-by-2 matrices act on variables \(x,y\), they also act on the space \(V=\Bbb C[x,y]\) of polynomials in \(x,y\). Namely, this action is given by the formulas \[e=x\partial_y,\;f=y\partial_x,\;h=x\partial_x-y\partial_y.\] This infinite-dimensional representation has the form \(V=\oplus_{n\ge 0}V_n\), where \(V_n\) is the space of polynomials of degree \(n\). The space \(V_n\) is invariant under \(e,f,h\), so it is an \(n+1\)-dimensional representation of \(\mathfrak{sl}_2\). It has basis \(v_{pq}=x^py^q\), such that \[hv_{pq}=(p-q)v_{pq},\;ev_{pq}=qv_{p+1,q-1},\;fv_{pq}=pv_{p-1,q+1}.\] Thus \(V_0\) is the trivial representation, and \(V_1\) is the tautological representation by 2-by-2 matrices. Also it is easy to see that \(V_2\) is the adjoint representation.

For a representation \(V\) define its character by \[\chi_V(z)={\rm Tr}_V(z^h)=\sum_m \dim V(m)z^m.\] Thus \[\chi_{V_n}(z)=z^n+z^{n-2}+...+z^{-n}=\frac{z^{n+1}-z^{-n-1}}{z-z^{-1}}.\] It is easy to see that \[\chi_{V\oplus W}=\chi_V+\chi_W, \chi_{V\otimes W}=\chi_V\chi_W.\] Since the functions \(\chi_{V_n}\) are linearly independent, we see that a finite dimensional representation of \(\mathfrak{sl}_2\) is determined by its character.

12 The universal enveloping algebra of a Lie algebra↩︎

12.1 The definition of the universal enveloping algebra↩︎

Let \(V\) be a vector space over a field \(\boldsymbol{k}\). Recall that the tensor algebra of \(V\) is the \(\Bbb Z\)-graded associative algebra \(TV:=\oplus_{n\ge 0}V^{\otimes n}\) (with \(\deg(V^{\otimes n})=n\)), with multiplication given by \(a\cdot b=a\otimes b\) for \(a\in V^{\otimes m}\) and \(b\in V^{\otimes n}\). If \(\lbrace x_i\rbrace\) is a basis of \(V\) then \(TV\) is just the free algebra with generators \(x_i\) (i.e., without any relations). Its basis consists of various words in the letters \(x_i\).

Let \(\mathfrak{g}\) be a Lie algebra over \({\boldsymbol{k}}\).

Recall that any associative algebra \(A\) is also a Lie algebra with operation \([a,b]:=ab-ba\). The following proposition follows immediately from the definition of \(U(\mathfrak{g})\).

Part (ii) of this proposition implies that any Lie algebra map
\(\psi: \mathfrak{g}\to A\) can be uniquely extended to an associative algebra map \(\phi: U(\mathfrak{g})\to A\) so that \(\psi=\phi\circ\rho\). This is the universal property of \(U(\mathfrak{g})\) which justifies the term “universal enveloping algebra".

In particular, it follows that a representation of \(\mathfrak{g}\) on a vector space \(V\) is the same thing as an algebra map \(U(\mathfrak{g})\to {\rm End}(V)\) (i.e., a representation of \(U(\mathfrak{g})\) on \(V\)). Thus, to understand the representation theory of \(\mathfrak{g}\), it is helpful to understand the structure of \(U(\mathfrak{g})\); for example, every central element \(C\in U(\mathfrak{g})\) gives rise to a morphism of representations \(V\to V\) (note that this has already come in handy in studying representations of \(\mathfrak{sl}_2\)).

In terms of the basis \(\lbrace x_i\rbrace\) of \(\mathfrak{g}\), we can write the bracket as \[[x_i,x_j]=\sum_k c_{ij}^kx_k,\] where \(c_{ij}^k\in {\boldsymbol{k}}\) are the structure constants. Then the algebra \(U(\mathfrak{g})\) can be described as the quotient of the free algebra \({\boldsymbol{k}}\langle \lbrace x_i\rbrace\rangle\) by the relations \[x_ix_j-x_jx_i=\sum_k c_{ij}^kx_k.\]

Recall that \(\mathfrak{g}\) acts on \(T\mathfrak{g}\) by derivations via the adjoint action. Moreover, using the Jacobi identity, we have \[{\rm ad}z(xy-yx-[x,y])=[z,x]y+x[z,y]-[z,y]x-y[z,x]-[z,[x,y]]=\] \[([z,x]y-y[z,x]-[[z,x],y])+(x[z,y]-[z,y]x-[x,[z,y]]).\] Thus \({\rm ad}z(I)\subset I\), and hence the action of \(\mathfrak{g}\) on \(T\mathfrak{g}\) descends to its action on \(U(\mathfrak{g})\) by derivations (also called the adjoint action). It is easy to see that these derivations are in fact inner: \[{\rm ad}z(a)=za-az\] for \(a\in U(\mathfrak{g})\) (although this is not so for \(T\mathfrak{g}\)). Indeed, it suffices to note that this holds for \(a\in \mathfrak{g}\) by the definition of \(U(\mathfrak{g})\).

Thus we get

12.2 Graded and filtered algebras↩︎

Recall that a \(\Bbb Z_{\ge 0}\)-filtered algebra is an algebra \(A\) equipped with a filtration \[0=F_{-1}A\subset F_0A\subset F_1A\subset...\subset F_nA\subset...\] such that \(1\in F_0A\), \(\cup_{n\ge 0} F_nA=A\) and \(F_iA\cdot F_jA\subset F_{i+j}A.\) In particular, if \(A\) is generated by \(\lbrace{x_\alpha\rbrace}\) then a filtration on \(A\) can be obtained by declaring \(x_\alpha\) to be of degree \(1\); i.e., \(F_nA=(F_1A)^n\) is the span of all words in \(x_\alpha\) of degree \(\le n\).

If \(A=\oplus_{i\ge 0}A_i\) is \(\Bbb Z_{\ge 0}\)-graded then we can define a filtration on \(A\) by setting \(F_nA:=\oplus_{i=0}^n A_i\); however, not any filtered algebra is obtained in this way, and having a filtration is a weaker condition than having a grading. Still, if \(A\) is a filtered algebra, we can define its associated graded algebra \({\rm gr}(A):=\oplus_{n\ge 0}{\rm gr}_n(A)\) (also denoted \({\rm gr}A\)), where \({\rm gr}_n(A):=F_nA/F_{n-1}A\). The multiplication in \({\rm gr}(A)\) is given by the “leading terms" of multiplication in \(A\): for \(a\in {\rm gr}_i(A),\;b\in {\rm gr}_j(A)\), pick their representatives \(\widetilde{a}\in F_iA,\;\widetilde{b}\in F_jA\) and let \(ab\) be the projection of \(\widetilde{a}\widetilde{b}\) to \({\rm gr}_{i+j}(A)\).

12.3 The coproduct of \(U(\mathfrak{g})\)↩︎

For a vector space \(\mathfrak{g}\) define the algebra homomorphism \(\Delta: T\mathfrak{g}\to T\mathfrak{g}\otimes T\mathfrak{g}\) given for \(x\in \mathfrak{g}\subset T\mathfrak{g}\) by \(\Delta(x)=x\otimes 1+1\otimes x\) (it exists and is unique since \(T\mathfrak{g}\) is freely generated by \(\mathfrak{g}\)).

The homomorphism \(\Delta\) is called the coproduct (of \(T\mathfrak{g}\) or \(U(\mathfrak{g})\)).

12.4 Differential operators on manifolds and Lie groups↩︎

We have seen in Subsection 5.2 that a vector field on a manifold \(X\) is the same thing as a derivation of the algebra \(O(U)\) for every open set \(U\subset X\) compatible with restriction maps \(O(U)\to O(V)\) for \(V\subset U\); in particular, on every \(U\) we have \([\boldsymbol{v},m_f]=m_{\boldsymbol{v}(f)}\) where \(f\in O(U)\) and \(m_f: O(U)\to O(U)\) is the operator of multiplication by \(f\in O(U)\). Thus if also \(g\in O(U)\) then \([[\boldsymbol{v},m_f],m_g]=0\). Conversely, if \(A\) is an endomorphism of the space \(O(U)\) for every open \(U\subset X\) compatible with restriction maps and \([[A,m_f],m_g]=0\) for any \(f,g\in O(U)\) then \(A=\boldsymbol{v}+m_h\) for a unique vector field \(\boldsymbol{v}\) and regular function \(h\) on \(X\) (check this!). This gives rise to the following generalization of the notion of a vector field.

It is easy to show that the latter condition is equivalent to the classical condition for a differential operator of order \(\le N\): in local coordinates \((x_i)\) on a chart \(U\subset X\) the operator \(A\) looks like \[A=\sum_{k=0}^N \sum_{i_1\le...\le i_k}F_{i_1,...,i_k}\frac{\partial^k}{\partial x_{i_1}...\partial x_{i_k}},\] where \(F_{i_1,...,i_k}\in O(U)\) (check this!). The space of such operators is denoted by \(D_N(X)\). Thus we have a nested sequence of spaces \[O(X)=D_0(X)\subset D_1(X)\subset...\subset D_N(X)\subset...\] The nested union \(\cup_{N\ge 0}D_N(X)\) is a filtered associative algebra called the algebra of differential operators on \(X\) and denoted by \(D(X)\).

Now suppose that a Lie group \(G\) with Lie algebra \(\mathfrak{g}\) acts on \(X\). Then we have a homomorphism of Lie algebras \(\mathfrak{g}\to {\rm Vect}(X)\), which can be viewed as a Lie algebra homomorphism \(\mathfrak{g}\to D(X)\). Thus by the universal property of the universal enveloping algebra, we obtain an associative algebra homomorphism \(\xi: U(\mathfrak{g})\to D(X)\). Moreover, this homomorphism preserves filtrations.

For example, if \(X=G\) and \(G\) acts by right translations, then the corresponding map \(\mathfrak{g}\to {\rm Vect}(G)\) identifies \(\mathfrak{g}\) with the Lie algebra \({\rm Vect}_L(G)\) of left-invariant vector fields on \(G\). Thus the map \(\xi: U(\mathfrak{g})\to D(G)\) lands in the subalgebra \(D_L(G)\) of left-invariant differential operators on \(G\).

13 The Poincaré-Birkhoff-Witt theorem↩︎

13.1 The statement of the Poincaré-Birkhoff-Witt theorem↩︎

Let \(\mathfrak{g}\) be a Lie algebra over a field \(\boldsymbol{k}\). Recall from Example 32 that we have a surjective algebra homomorphism \[\phi: S\mathfrak{g}\to {\rm gr}U(\mathfrak{g}).\]

We will prove Theorem 78 in Subsection 13.2. Now let us discuss its reformulation in terms of a basis and corollaries.

Given a basis \(\lbrace x_i\rbrace\) of \(\mathfrak{g}\), fix an ordering on this basis and consider ordered monomials \(\prod_i x_i^{n_i}\), where the product is ordered according to the ordering of the basis. The statement that \(\phi\) is surjective is equivalent to saying that ordered monomials span \(U(\mathfrak{g})\). This is also easy to see directly: any monomial can be ordered using the commutation relations at the cost of an error of lower degree, so proceeding recursively, we can write any monomial as a linear combination of ordered ones. Thus the PBW theorem can be formulated as follows:

For instance, if \(\boldsymbol{k}=\Bbb R\) or \(\Bbb C\) and \(\mathfrak{g}={\rm Lie}(G)\) where \(G\) is a Lie group, this theorem is easy to deduce from Exercise 77 (do this!).

Let \({\rm char}(\boldsymbol{k})=0\). Define the symmetrization map \(\sigma: S\mathfrak{g}\to U(\mathfrak{g})\) given by \[\sigma(y_1\otimes...\otimes y_n)=\frac{1}{n!}\sum_{s\in S_n}y_{s(1)}...y_{s(n)}.\] It is easy to see that this map commutes with the adjoint action of \(\mathfrak{g}\).

Let \(Z(U(\mathfrak{g}))\) denote the center of \(U(\mathfrak{g})\).

In the case when \(\mathfrak{g}={\rm Lie}G\) for a connected Lie group \(G\), we thus obtain a filtered vector space isomorphism of the center of \(U(\mathfrak{g})\) with \((S\mathfrak{g})^{{\rm Ad} G}\).

13.2 Proof of the PBW theorem↩︎

The proof of Theorem 78 is based on the following key lemma.

Note that Lemma 8 immediately implies the PBW theorem, since by this lemma the images of ordered monomials under \(\varphi\) are linearly independent in \(S\mathfrak{g}\), implying that these monomials themselves are linearly independent in \(U(\mathfrak{g})\).

14 Free Lie algebras, the Baker-Campbell-Hausdorff formula↩︎

14.1 Primitive elements↩︎

Let \(\mathfrak{g}\) be a Lie algebra over a field \(\boldsymbol{k}\). Let us say that \(x\in U(\mathfrak{g})\) is primitive if \(\Delta(x)=x\otimes 1+1\otimes x\). It is clear that if \(x\in\mathfrak{g}\subset U(\mathfrak{g})\) then \(x\) is primitive.

14.2 Free Lie algebras↩︎

Let \(V\) be a vector space over a field \(\boldsymbol{k}\). The free Lie algebra \(L(V)\) generated by \(V\) is the Lie subalgebra of \(TV\) generated by \(V\). Note that \(L(V)\) is a \(\Bbb Z_{>0}\)-graded Lie algebra: \(L(V)=\oplus_{m\ge 1}L_m(V)\), with grading defined by \(\deg V=1\); thus \(L_m(V)\) is spanned by commutators of \(m\)-tuples of elements of \(V\) inside \(TV\).

The Lie algebra embedding \(L(V)\hookrightarrow TV\) gives rise to an associative algebra homomorphism \(\psi: U(L(V))\to TV\).

14.3 The Baker-Campbell-Hausdorff formula↩︎

We have defined the commutator \([x,y]\) on \(\mathfrak{g}={\rm Lie}G\) as the quadratic part of \(\mu(x,y)=\log(\exp(x)\exp(y))\). So one may wonder if taking higher order terms in the Taylor explansion of \(\mu(x,y)\), \[\label{CHser} \mu(x,y)\sim \sum_{n=1}^\infty\frac{\mu_n(x,y)}{n!}\tag{5}\] would yield new operations on \(\mathfrak{g}\). It turns out, however, that all these operations express via the commutator. Namely, we have

15 Solvable and nilpotent Lie algebras, theorems of Lie and Engel↩︎

15.1 Ideals and commutant↩︎

Let \(\mathfrak{g}\) be a Lie algebra. Recall that an ideal in \(\mathfrak{g}\) is a subspace \(\mathfrak{h}\) such that \([\mathfrak{g},\mathfrak{h}]\subset \mathfrak{h}\). If \(\mathfrak{h}\subset \mathfrak{g}\) is an ideal then \(\mathfrak{g}/\mathfrak{h}\) has a natural structure of a Lie algebra. Moreover, if \(\phi: \mathfrak{g}_1\to \mathfrak{g}_2\) is a homomorphism of Lie algebras then \({\rm Ker}\phi\) is an ideal in \(\mathfrak{g}_1\), \({\rm Im}\phi\) is a Lie subalgebra in \(\mathfrak{g}_2\), and \(\phi\) induces an isomorphism \(\mathfrak{g}_1/{\rm Ker}\phi\cong {\rm Im}\phi\) (check it!).

15.2 Solvable Lie algebras↩︎

For a Lie algebra \(\mathfrak{g}\) define its derived series recursively by the formulas \(D^0(\mathfrak{g})=\mathfrak{g}\), \(D^{n+1}(\mathfrak{g})=[D^n(\mathfrak{g}),D^n(\mathfrak{g})]\). This is a descending sequence of ideals in \(\mathfrak{g}\).

15.3 Nilpotent Lie algebras↩︎

For a Lie algebra \(\mathfrak{g}\) define its lower central series recursively by the formulas \(D_0(\mathfrak{g})=\mathfrak{g}\), \(D_{n+1}(\mathfrak{g})=[\mathfrak{g},D_n(\mathfrak{g})]\). This is a descending sequence of ideals in \(\mathfrak{g}\).

15.4 Lie’s theorem↩︎

One of the main technical tools of the structure theory of finite dimensional Lie algebras is Lie’s theorem for solvable Lie algebras. Before stating and proving this theorem, we will prove the following auxiliary lemma, which will be used several times.

Here is another formulation of Lie’s theorem:

In the case \(\dim\mathfrak{g}=1\), this recovers the well known theorem in linear algebra that any linear operator on a finite dimensional \({\boldsymbol{k}}\)-vector space is upper triangular in some basis (which is actually true in any characteristic).

15.5 Engel’s theorem↩︎

Another key tool of the structure theory of finite dimensional Lie algebras is Engel’s theorem. Before stating and proving this theorem, we prove an auxiliary result.

16 Semisimple and reductive Lie algebras, the Cartan criteria↩︎

16.1 Semisimple and reductive Lie algebras, the radical↩︎

Let \(\mathfrak{g}\) be a finite dimensional Lie algebra over a field \(\mathbf{k}\).

Thus if \(\mathfrak{g}\) is both solvable and semisimple then \(\mathfrak{g}=0\).

In fact, in characteristic zero there is a stronger statement, which says that the extension in Theorem 104 splits. Namely, given a Lie algebra \(\mathfrak{h}\) and another Lie algebra \(\mathfrak{a}\) acting on \(\mathfrak{h}\) by derivations, we may form the semidirect product Lie algebra \(\mathfrak{a}\ltimes \mathfrak{h}\) which is \(\mathfrak{a}\oplus \mathfrak{h}\) as a vector space with commutator defined by \[[(a_1,h_1),(a_2,h_2)]=([a_1,a_2],a_1\circ h_2-a_2\circ h_1+[h_1,h_2]).\] Note that a special case of this construction has already appeared in Example 39.

Theorem 105 will be proved in Subsection 48.2.

In other words, \(\mathfrak{g}\) is reductive if \([\mathfrak{g},{\rm rad}(\mathfrak{g})]=0\).

The Levi decomposition theorem implies that a reductive Lie algebra in characteristic zero is a direct sum of a semisimple Lie algebra and an abelian Lie algebra (its center). We will also prove this in Corollary 30.

16.2 Invariant inner products↩︎

Let \(B\) be a bilinear form on a Lie algebra \(\mathfrak{g}\). Recall that \(B\) is invariant if \(B([x,y],z)=B(x,[y,z])\) for any \(x,y,z\in \mathfrak{g}\).

In fact, we have the following proposition.

For example, the Lie algebras \(\mathfrak{so}_n(\Bbb K)\), \(\mathfrak{sp}_{2n}(\Bbb K)\), \({\mathfrak{su}}(p,q)\) have trivial center and therefore are semisimple.

16.3 The Killing form and the Cartan criteria↩︎

The Killing form is denoted by \(K_\mathfrak{g}(x,y)\) or shortly by \(K(x,y)\).

Theorems 111 and 112 will be proved in the next section.

16.4 Jordan decomposition↩︎

To prove the Cartan criteria, we will use the Jordan decomposition of a square matrix. Let us recall it.

17 Proofs of the Cartan criteria, properties of semisimple Lie algebras↩︎

17.1 Proof of the Cartan solvability criterion↩︎

It is clear that \(\mathfrak{g}\) is solvable if and only if so is \(\mathfrak{g}\otimes_{\boldsymbol{k}}\overline{\boldsymbol{k}}\), so we may assume that \({\boldsymbol{k}}\) is algebraically closed.

For the “only if" part, note that by Lie’s theorem, \(\mathfrak{g}\) has a basis in which the operators \({\rm ad}x\), \(x\in \mathfrak{g}\), are upper triangular. Then \([\mathfrak{g},\mathfrak{g}]\) acts in this basis by strictly upper triangular matrices, so \(K(x,y)=0\) for \(x\in [\mathfrak{g},\mathfrak{g}]\) and \(y\in \mathfrak{g}\).

To prove the “if" part, let us prove the following lemma.

Now the “if" part of the Cartan solvability criterion follows easily by applying Lemma 13 to the faithful representation \(V=\mathfrak{g}\) of the Lie algebra \(\mathfrak{g}/\mathfrak{z}(\mathfrak{g})\).

17.2 Proof of the Cartan semisimplicity criterion↩︎

Assume that \(\mathfrak{g}\) is semisimple, and let \(I={\rm Ker}(K_\mathfrak{g})\), an ideal in \(\mathfrak{g}\). Then \(K_I=(K_\mathfrak{g})|_I=0\). Thus by Cartan’s solvability criterion \(I\) is solvable. Hence \(I=0\).

Conversely, suppose \(K_\mathfrak{g}\) is nondegenerate. Then \(\mathfrak{g}\) is reductive. Moreover, the center of \(\mathfrak{g}\) is contained in the kernel of \(K_\mathfrak{g}\), so it must be trivial. Thus \(\mathfrak{g}\) is semisimple.

17.3 Properties of semisimple Lie algebras↩︎

We will see below (in Proposition 118) that \(J\) is in fact unique and must equal \(I^\perp\).

Let \({\rm Der}\mathfrak{g}\) be the Lie algebra of derivations of a Lie algebra \(\mathfrak{g}\). We have a homomorphism \({\rm ad}: \mathfrak{g}\to {\rm Der}\mathfrak{g}\) whose kernel is the center \(\mathfrak{z}(\mathfrak{g})\). Thus if \(\mathfrak{g}\) has trivial center (e.g., is semisimple) then the map \({\rm ad}\) is injective and identifies \(\mathfrak{g}\) with a Lie subalgebra of \({\rm Der}\mathfrak{g}\). Moreover, for \(d\in {\rm Der}\mathfrak{g}\) and \(x\in \mathfrak{g}\), we have \[[d,{\rm ad}x](y)=d[x,y]-[x,dy]=[dx,y]={\rm ad}(dx)(y).\] Thus \(\mathfrak{g}\subset {\rm Der}\mathfrak{g}\) is an ideal.

18 Extensions of representations, Whitehead’s theorem, complete reducibility↩︎

18.1 Extensions↩︎

Let \(\mathfrak{g}\) be a Lie algebra and \(U,W\) be representations of \(\mathfrak{g}\). We would like to classify all representations \(V\) which fit into a short exact sequence \[\label{seseq} 0\to U\to V\to W\to 0,\tag{8}\] i.e., \(U\subset V\) is a subrepresentation such that the surjection \(p: V\to W\) has kernel \(U\) and thus defines an isomorphism \(V/U\cong W\). In other words, \(V\) is endowed with a 2-step filtration with \(F_0V=U\) and \(F_1V=V\) such that \(F_1V/F_0V=W\), so \({\rm gr}(V)=U\oplus W\). To do so, pick a splitting of this sequence as a sequence of vector spaces, i.e. an injection \(i: W\to V\) (not a homomorphism of representations, in general) such that \(p\circ i={\rm Id}_W\). This defines a linear isomorphism \(\widetilde{i}: U\oplus W\to V\) given by \((u,w)\mapsto u+i(w)\), which allows us to rewrite the action of \(\mathfrak{g}\) on \(V\) as an action on \(U\oplus W\). Since \(\widetilde{i}\) is not in general a morphism of representations, this action is given by \[\rho(x)(u,w)=(xu+a(x)w,xw)\] where \(a: \mathfrak{g}\to {\rm Hom}_{\boldsymbol{k}}(W,U)\) is a linear map, and \(\widetilde{i}\) is a morphism of representations iff \(a=0\).

What are the conditions on \(a\) to give rise to a representation? We compute: \[\rho([x,y])(u,w)= ([x,y]u+a([x,y])w,[x,y]w),\] \[[\rho(x),\rho(y)](u,w)=([x,y]u+([x,a(y)]+[a(x),y])w,[x,y]w).\] Thus the condition to give a representation is the Leibniz rule \[a([x,y])=[x,a(y)]+[a(x),y]=[x,a(y)]-[y,a(x)].\] In general, if \(E\) is a representation of \(\mathfrak{g}\) then a linear function \(a: \mathfrak{g}\to E\) such that \[a([x,y])=x\circ a(y)-y\circ a(x)\] is called a \({\boldsymbol{1}-cocycle}\) of \(\mathfrak{g}\) with values in \(E\). The space of 1-cocycles is denoted by \(Z^1(\mathfrak{g},E)\).

Thus we see that in our setting \(a: \mathfrak{g}\to {\rm Hom}_{\boldsymbol{k}}(W,U)\) defines a representation if and only if \(a\in Z^1(\mathfrak{g}, {\rm Hom}_{\boldsymbol{k}}(W,U))\). Denote the representation \(V\) attached to such \(a\) by \(V_a\). Then we have a natural short exact sequence \[0\to U\to V_a\to W\to 0.\] It may, however, happen that some \(a\ne 0\) defines a trivial extension \(V\cong U\oplus W\), i.e., \(V_a\cong V_0\), and more generally \(V_a\cong V_b\) for \(a\ne b\). Let us determine when this happens. More precisely, let us look for isomorphisms \(f: V_a\to V_b\) preserving the structure of the short exact sequences, i.e., such that \({\rm gr}(f)={\rm Id}\). Then \[f(u,w)=(u+Aw,w)\] where \(A: W\to U\) is a linear map. Then we have \[xf(u,w)=x(u+Aw,w)=(xu+xAw+b(x)w,xw)\] and \[fx(u,w)=f(xu+a(x)w,xw)=(xu+a(x)w+Axw,xw),\] so we get that \(xf=fx\) iff \[[x,A]=a(x)-b(x).\] In particular, setting \(b=0\), we see that \(V\) is a trivial extension if and only if \(a(x)=[x,A]\) for some \(A\).

More generally, if \(E\) is a \(\mathfrak{g}\)-module, the linear function \(a: \mathfrak{g}\to E\) given by \(a(x)=xv\) for some \(v\in E\) is called the 1-coboundary of \(v\), and one writes \(a=dv\). The space of 1-coboundaries is denoted by \(B^1(\mathfrak{g},E)\); it is easy to see that it is a subspace of \(Z^1(\mathfrak{g},E)\), i.e., a 1-coboundary is always a 1-cocycle. Thus in our setting \(f: V_a\to V_b\) is an isomorphism of representations iff \[a-b=dA,\] i.e., there is an isomorphism \(f: V_a\cong V_b\) with \({\rm gr}(f)={\rm Id}\) if and only if \(a=b\) in the quotient space \[{\rm Ext}^1(W,U):=Z^1(\mathfrak{g},{\rm Hom}_{\boldsymbol{k}}(W,U))/B^1(\mathfrak{g},{\rm Hom}_{\boldsymbol{k}}(W,U)).\] The notation is justified by the fact that this space parametrizes extensions of \(W\) by \(U\). More precisely, every short exact sequence 8 gives rise to a class \([V]\in {\rm Ext}^1(W,U)\), and the extension defined by this sequence is trivial iff \([V]=0\).

More generally, for a \(\mathfrak{g}\)-module \(E\) the space \[H^1(\mathfrak{g},E):=Z^1(\mathfrak{g},E)/B^1(\mathfrak{g},E)\] is called the first cohomology of \(\mathfrak{g}\) with coefficients in \(E\). Thus, \[\mathop{\mathrm{Ext}}\nolimits^1(W,U)=H^1(\mathfrak{g}, {\rm Hom}_{\boldsymbol{k}}(W,U)).\]

18.2 Whitehead’s theorem↩︎

We have shown in Corollary 26 and Proposition 119 that for a semisimple \(\mathfrak{g}\) over a field of characteristic zero, \(H^1(\mathfrak{g},{\boldsymbol{k}})=(\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*=0\), and \(H^1(\mathfrak{g},\mathfrak{g})={\rm Der}\mathfrak{g}/\mathfrak{g}=0\). In fact, these are special cases of a more general theorem.

18.3 Proof of Theorem 121↩︎

By extension of scalars, we may assume without loss of generality that \(\mathbf{k}\) is algebraically closed. We will use the following lemma, which actually holds over any field.

It remains to construct a central element of \(U(\mathfrak{g})\) for a semisimple Lie algebra \(\mathfrak{g}\) to which we can apply Lemma 15. This can be done as follows. Let \(a_i\) be a basis of \(\mathfrak{g}\) and \(a^i\) the dual basis under an invariant inner product on \(\mathfrak{g}\) (for example, the Killing form). Define the (quadratic) Casimir element \[C:=\sum_i a_ia^i.\] It is easy to show that \(C\) is independent of the choice of the basis (although it depends on the choice of the inner product). Also \(C\) is central: for \(y\in \mathfrak{g}\), \[[y,C]=\sum_i ([y,a_i]a^i+a_i[y,a^i])=0\] since \[\sum_i ([y,a_i]\otimes a^i+a_i\otimes [y,a^i])=0\] (this is seen by taking the inner product of the first tensorand with \(a^j\) and using the invariance of the inner product). Finally, note that for \(\mathfrak{g}=\mathfrak{sl}_2\), \(C\) is proportional to the Casimir element \(2fe+\frac{h^2}{2}+h=ef+fe+\frac{h^2}{2}\) considered previously, as the basis \(f,e,\frac{h}{\sqrt{2}}\) is dual to the basis \(e,f,\frac{h}{\sqrt{2}}\) under an invariant inner product of \(\mathfrak{g}\).

The key lemma used in the proof of Theorem 121 is the following.

Now we can prove Theorem 121. By Lemma 14, it suffices to prove the theorem for irreducible \(V\), which is guaranteed by Corollary 29.

18.4 Complete reducibility of representations of semisimple Lie algebras↩︎

19 Structure of semisimple Lie algebras, I↩︎

19.1 Semisimple elements↩︎

Let \({\boldsymbol{k}}\) be an algebraically closed field, and \(\mathfrak{g}\) be a finite dimensional Lie algebra over \({\boldsymbol{k}}\). Let \(x\in \mathfrak{g}\). Let \(\mathfrak{g}_\lambda\subset \mathfrak{g}\) be the generalized eigenspace of \({\rm ad}x\) with eigenvalue \(\lambda\). Then \(\mathfrak{g}=\oplus_\lambda \mathfrak{g}_\lambda\).

It is clear that any element which is both semisimple and nilpotent is central, so for a semisimple Lie algebra it must be zero. Note also that for \(\mathfrak{g}=\mathfrak{sl}_n({\boldsymbol{k}})\) this coincides with the usual definition.

19.2 Toral subalgebras↩︎

From now on we assume that \({\rm char}(\boldsymbol{k})=0\) unless specified otherwise.

19.3 Cartan subalgebras↩︎

It is clear that any Cartan subalgebra is a maximal toral subalgebra of \(\mathfrak{g}\). The following theorem, stating the converse, shows that Cartan subalgebras exist.

We will show in Theorem 132 that all Cartan subalgebras of \(\mathfrak{g}\) are conjugate under \({\rm Aut}(\mathfrak{g})\), in particular they all have the same dimension, which is called the rank of \(\mathfrak{g}\).

19.4 Root decomposition↩︎

Now let \(\mathfrak{g}\) be a semisimple Lie algebra and \(\mathfrak{h}\subset \mathfrak{g}\) a Cartan subalgebra. Let \((,)\) be a nondegenerate invariant symmetric bilinear form on \(\mathfrak{g}\), for example the Killing form. Since the restriction of \((,)\) to \(\mathfrak{h}\) is nondegenerate, it defines an isomorphism \(\mathfrak{h}\to \mathfrak{h}^*\) given by \(h\mapsto (h,?)\). The inverse of this isomorphism will be denoted by \(\alpha\mapsto H_\alpha\). We also have the inverse form on \(\mathfrak{h}^*\) which we also will denote by \((,)\); it is given by \((\alpha,\beta):=\alpha(H_\beta)=(H_\alpha,H_\beta)\).

The Lie subalgebra of \(\mathfrak{g}\) spanned by \(e,f,h_\alpha\), which we’ve shown to be isomorphic to \({\mathfrak{sl}}_2(\boldsymbol{k})\), will be denoted by \({\mathfrak{sl}}_2(\boldsymbol{k})_\alpha\) (we will see that \(\mathfrak{g}_\alpha\) are 1-dimensional so it is independent of the choices).

20 Structure of semisimple Lie algebras, II↩︎

20.1 Strongly regular (regular semisimple) elements↩︎

In this section we will discuss another way of constructing Cartan subalgebras. First consider an example.

So we may hope that if we take a generic element \(x\) in a semisimple Lie algebra then its centralizer is a Cartan subalgebra. But for that we have to define what we mean by generic.

We will need the following auxiliary lemma.

20.2 Conjugacy of Cartan subalgebras↩︎

We note that there is also a useful notion of a regular element, which is an \(x\in \mathfrak{g}\) for which the ordinary (rather than generalized) \(0\)-eigenspace of \({\rm ad}x\) (i.e., the centralizer \(C(x)\) of \(x\)) has dimension \({\rm rank}\mathfrak{g}\). Such elements don’t have to be semisimple, e.g. the nilpotent Jordan block in \(\mathfrak{sl}_n\) is regular. It follows from Corollary 35(i) that an element is strongly regular if and only if it is both regular and semisimple. For this reason, from now on we will follow standard terminology and call strongly regular elements regular semisimple.

20.3 Root systems of classical Lie algebras↩︎

21 Root systems↩︎

21.1 Abstract root systems↩︎

Let \(E\cong \Bbb R^r\) be a Euclidean space with a positive definite inner product.

In particular, taking \(\beta=\alpha\) in R3 yields that \(R\) is centrally symmetric, i.e., \(R=-R\). Also note that \(s_\alpha\) is the reflection with respect to the hyperplane \((\alpha,x)=0\), so R3 just says that \(R\) is invariant under such reflections.

Note also that if \(R\subset E\) is a root system, \(\overline{E}\subset E\) a subspace, and \(R'=R\cap \overline{E}\) then \(R'\) is also a root system inside \(E'={\rm Span}(R')\subset \overline{E}\).

For a root \(\alpha\) the corresponding coroot \(\alpha^\vee\in E^*\) is defined by the formula \(\alpha^\vee(x)=\frac{2(\alpha,x)}{(\alpha,\alpha)}\). Thus \(\alpha^\vee(\alpha)=2\), \(n_{\alpha\beta}=\alpha^\vee(\beta)\) and \(s_\alpha(\beta)=\beta-\alpha^\vee(\beta)\alpha\).

So an isomorphism does not have to preserve the inner product, e.g. it may rescale it.

21.2 The Weyl group↩︎

21.3 Root systems of rank \(2\)↩︎

If \(\alpha,\beta\) are linearly independent roots in \(R\) and \(E'\subset E\) is spanned by \(\alpha,\beta\) then \(R'=R\cap E'\) is a root system in \(E'\) of rank \(2\). So to classify reduced root systems, it is important to classify reduced root systems of rank \(2\) first.

In fact, all these possibilities are realized. Namely, we have root systems \(A_1\times A_1\), \(A_2\), \(B_2=C_2\) (the root system of the Lie algebras \(\mathfrak{sp}_4\) and \(\mathfrak{so}_5\), which are in fact isomorphic, consisting of the vertices and midpoints of edges of a square), and \(G_2\), generated by \(\alpha,\beta\) with \((\alpha,\alpha)=6\), \((\beta,\beta)=2\), \((\alpha,\beta)=-3\), and roots being \(\pm \alpha,\pm \beta\), \(\pm (\alpha+\beta)\), \(\pm (\alpha+2\beta)\), \(\pm (\alpha+3\beta)\), \(\pm (2\alpha+3\beta)\).

The root systems of rank \(2\) are shown in the following picture.

21.4 Positive and simple roots↩︎

Let \(R\) be a reduced root system and \(t\in E^*\) be such that \(t(\alpha)\ne 0\) for any \(\alpha\in R\). We say that a root is positive (with respect to \(t\)) if \(t(\alpha)>0\) and negative if \(t(\alpha)<0\). The set of positive roots is denoted by \(R_+\) and of negative ones by \(R_-\), so \(R_+=-R_-\) and \(R=R_+\cup R_-\) (disjoint union). This decomposition is called a polarization of \(R\); it depends on the choice of \(t\).

Thus the set \(\Pi\) of simple roots has \(r\) elements: \(\Pi=(\alpha_1,...,\alpha_r)\).

We thus obtain

For a positive root \(\alpha\), its height \(h(\alpha)\) is the number \(\sum n_i\). So simple roots are the roots of height \(1\), and the height of \({\boldsymbol{e}}_i-{\boldsymbol{e}}_j\) in \(R=A_{n-1}\) is \(j-i\).

21.5 Dual root system↩︎

For a root system \(R\), the set \(R^\vee\subset E^*\) of \(\alpha^\vee\) for all \(\alpha\in R\) is also a root system, such that \((R^\vee)^\vee=R\). It is called the dual root system to \(R\). For example, \(B_n\) is dual to \(C_n\), while \(A_{n-1}\), \(D_n\) and \(G_2\) are self-dual.

Moreover, it is easy to see that any polarization of \(R\) gives rise to a polarization of \(R^\vee\) (using the image \(t^\vee\) of \(t\) under the isomorphism \(E\to E^*\) induced by the inner product), and the corresponding system \(\Pi^\vee\) of simple roots consists of \(\alpha_i^\vee\) for \(\alpha_i\in \Pi\).

21.6 Root and weight lattices↩︎

Recall that a lattice in a real vector space \(E\) is a subgroup \(Q\subset E\) generated by a basis of \(E\). Of course, every lattice is conjugate to \(\Bbb Z^n\subset \Bbb R^n\) by an element of \(GL_n(\Bbb R)\). Also recall that for a lattice \(Q\subset E\) the dual lattice \(Q^*\subset E^*\) is the set of \(f\in E^*\) such that \(f(v)\in \Bbb Z\) for all \(v\in Q\). If \(Q\) is generated by a basis \({\boldsymbol{e}}_i\) of \(E\) then \(Q^*\) is generated by the dual basis \({\boldsymbol{e}}_i^*\).

In particular, for a root system \(R\) we can define the root lattice \(Q\subset E\), which is generated by the simple roots \(\alpha_i\) with respect to some polarization of \(R\). Since \(Q\) is also generated by all roots in \(R\), it is independent of the choice of the polarization. Similarly, we can define the coroot lattice \(Q^\vee\subset E^*\) generated by \(\alpha^\vee,\alpha\in R\), which is just the root lattice of \(R^\vee\).

Also we define the weight lattice \(P\subset E\) to be the dual lattice to \(Q^\vee\): \(P=(Q^\vee)^*\), and the coweight lattice \(P^\vee\subset E^*\) to be the dual lattice to \(Q\): \(P^\vee=Q^*\), so \(P^\vee\) is the weight lattice of \(R^\vee\). Thus \[P=\lbrace \lambda\in E: (\lambda,\alpha^\vee)\in \Bbb Z\;\forall \alpha\in R\rbrace,\;P^\vee=\lbrace \lambda\in E^*: (\lambda,\alpha)\in \Bbb Z\;\forall \alpha\in R\rbrace.\]

Since for \(\alpha,\beta\in R\) we have \((\alpha^\vee,\beta)=n_{\alpha\beta}\in \Bbb Z\), we have \(Q\subset P\), \(Q^\vee\subset P^\vee\).

Given a system of simple roots \(\Pi=\lbrace \alpha_1,...,\alpha_r\rbrace\), we define fundamental coweights \(\omega_i^\vee\) to be the dual basis to \(\alpha_i\) and fundamental weights \(\omega_i\) to be the dual basis to \(\alpha_i^\vee\): \((\omega_i,\alpha_j^\vee)=(\omega_i^\vee,\alpha_j)=\delta_{ij}\). Thus \(P\) is generated by \(\omega_i\) and \(P^\vee\) by \(\omega_i^\vee\).

22 Properties of the Weyl group↩︎

22.1 Weyl chambers↩︎

Suppose we have two polarizations of a root system \(R\) defined by \(t,t'\in E^*\), and \(\Pi,\Pi'\) are the corresponding systems of simple roots. Are \(\Pi,\Pi'\) equivalent in a suitable sense? The answer turns out to be yes. To show this, we will need the notion of a Weyl chamber.

Note that the polarization defined by \(t\) depends only on the signs of \(t(\alpha)\), so does not change when \(t\) is continuously deformed without crossing the hyperplanes \(t(\alpha)=0\). This motivates the following definition:

Thus a Weyl chamber is defined by a system of strict homogeneous linear inequalities \(\pm (\alpha,x)=0\), \(\alpha\in R\). More precisely, the set of solutions of such a system is either empty or a Weyl chamber.

Thus the polarization defined by \(t\) depends only on the Weyl chamber containing \(t\).

The following lemma is geometrically obvious.

The root hyperplanes containing the faces \(F_i\) are called the walls of \(C\).

We have seen above that every Weyl chamber defines a polarization of \(R\). Conversely, every polarization defines the corresponding positive Weyl chamber \(C_+\) defined by the conditions \((\alpha,x)>0\) for \(\alpha\in R_+\) (this set is nonempty since it contains \(t\), hence is a Weyl chamber). Thus \(C_+\) is the set of vectors of the form \(\sum_{i=1}^r c_i\omega_i\) with \(c_i>0\). So \(C_+\) has \(r\) faces \(L_{\alpha_1}\cap \overline{C}_+,...,L_{\alpha_r}\cap \overline{C}_+\).

Since the Weyl group \(W\) permutes the roots, it acts on the set of Weyl chambers.

22.2 Simple reflections↩︎

Given a polarization of \(R\) and the corresponding system of simple roots \(\Pi=\lbrace \alpha_1,...,\alpha_r\rbrace\), the simple reflections are the reflections \(s_{\alpha_i}\), denoted by \(s_i\).

Thus \(R\) can be reconstructed from \(\Pi\) as \(W(\Pi)\), where \(W\) is the subgroup of \(O(E)\) generated by \(s_i\).

22.3 Length of an element of the Weyl group↩︎

Let us say that a root hyperplane \(L_\alpha\) separates two Weyl chambers \(C,C'\) if they lie on different sides of \(L_\alpha\).

We have \(t\in C_+, w(t)\in w(C_+)\), so \(\ell(w)\) is the number of roots \(\alpha\) such that \((t,\alpha)>0\) but \((w(t),\alpha)=(t,w^{-1}\alpha)<0\). Note that if \(\alpha\) is a root satisfying this condition then \(\beta=-w^{-1}\alpha\) satisfies the conditions \((t,\beta)>0\), \((t,w\beta)<0\). Thus \(\ell(w)=\ell(w^{-1})\) and \(\ell(w)\) is the number of positive roots which are mapped by \(w\) to negative roots. Note also that the notion of length depends on the polarization of \(R\) (as it refers to the positive chamber \(C_+\) defined using the polarization).

The weight \(\rho\) plays an important role in representation theory of semisimple Lie algebras. For instance, it occurs in the Weyl character formula for these representations which we will soon derive.

An expression \(w=s_{i_1}...s_{i_l}\) is called reduced if \(l=\ell(w)\).

Thus we see that \(\overline{C}_+\) is a fundamental domain of the action of \(W\) on \(E\).

Moreover, we have

The element \(w_0\) is therefore called the longest element of \(W\).

23 Dynkin diagrams↩︎

23.1 Cartan matrices and Dynkin diagrams↩︎

Our goal now is to classify reduced root systems, which is a key step in the classification of semisimple Lie algebras. We have shown that classifying root systems is equivalent to classifying sets \(\Pi\) of simple roots. So we need to classify such sets \(\Pi\). Before doing so, note that we have a nice notion of direct product of root systems.

Namely, let \(R_1\subset E_1\) and \(R_2\subset E_2\) be two root systems. Let \(E=E_1\oplus E_2\) (orthogonal decomposition) and \(R=R_1\sqcup R_2\) (with \(R_1\perp R_2\)). If \(t_1\in E_1,t_2\in E_2\) define polarizations of \(R_1,R_2\) with systems of simple roots \(\Pi_1,\Pi_2\) then \(t=t_1+t_2\) defines a polarization of \(R\) with \(\Pi=\Pi_1\sqcup \Pi_2\) (with \(\Pi_1\perp \Pi_2\) and \(\Pi_i=\Pi\cap R_i\)).

Thus it suffices to classify irreducible root systems.

As noted above, a root system is determined by pairwise inner products of positive roots. However, it is more convenient to encode them by the Cartan matrix \(A\) defined by \[a_{ij}=n_{\alpha_j\alpha_i}=(\alpha_i^\vee,\alpha_j).\] The following properties of the Cartan matrix follow immediately from Lemma 24, Theorem 136 and Theorem 138:

We will see later that conversely, any such matrix defines a root system.

It is convenient to encode such matrices by Dynkin diagrams:

\(\bullet\) Indices \(i\) are vertices;

\(\bullet\) Vertices \(i\) and \(j\) are connected by \(a_{ij}a_{ji}\) lines;

\(\bullet\) If \(a_{ij}\ne a_{ji}\), i.e., \(|\alpha_i|^2\ne |\alpha_j|^2\), then the arrow on the lines goes from long root to short root (“less than" sign).

It is clear that such a diagram completely determines the Cartan matrix (if we fix the labeling of vertices), and vice versa. Also it is clear that the root system is irreducible if and only if its Dynkin diagram is connected.

23.2 Classification of Dynkin diagrams↩︎

The following theorem gives a complete classification of irreducible root systems.

The proof of Theorem 149 is rather long but direct. It consists of several steps. The first step is construction of the remaining root systems \(F_4,E_6,E_7,E_8\).

23.3 The root system \(F_4\)↩︎

Thus besides the roots of \(B_4\), which are \(\pm {\boldsymbol{e}}_i\pm {\boldsymbol{e}}_j\) (24 of them, squared length \(2\)) and \(\pm {\boldsymbol{e}}_i\) (8 of them, squared length \(1\)), we have the 16 new roots \(\sum_{i=1}^4 (\pm \frac{1}{2}{\boldsymbol{e}}_i)\) (squared length \(1\)); this gives a total of 48.

To give a polarization of the \(F_4\) root system, pick \(t=(t_1,t_2,t_3,t_4)\) with \(t_1\gg t_2\gg t_3\gg t_4\).

23.4 The root system \(E_8\)↩︎

Thus besides the roots of \(D_8\), \(\pm {\boldsymbol{e}}_i\pm {\boldsymbol{e}}_j\) (112 of them), we have 128 new roots \(\sum_{i=1}^8 (\pm \frac{1}{2}{\boldsymbol{e}}_i)\). So in total we have \(240\) roots. All roots have squared length \(2\).

To give a polarization of the \(E_8\) root system, pick \(t\) so that \(t_i\gg t_{i+1}\).

23.5 The root system \(E_7\)↩︎

Note that these roots (unlike \(\alpha_8={\boldsymbol{e}}_2-{\boldsymbol{e}}_3\)) satisfy the equation \(x_1+x_2=0\). Thus \(E_7\) is the intersection of \(E_8\) with this subspace. So it includes the roots \(\pm {\boldsymbol{e}}_i\pm {\boldsymbol{e}}_j\) with \(3\le i,j\le 8\) distinct (60 roots), \(\pm ({\boldsymbol{e}}_1-{\boldsymbol{e}}_2)\) (2 roots) and \(\sum_{i=1}^8 (\pm \frac{1}{2}{\boldsymbol{e}}_i)\) with even number of minuses and the opposite signs for \({\boldsymbol{e}}_1\) and \({\boldsymbol{e}}_2\) (64 roots). Altogether we get 126 roots. The Cartan matrix is the upper left corner 7 by 7 submatrix of the Cartan matrix of \(E_8\), so it is \[A=\begin{pmatrix} 2& 0 & -1& 0& 0& 0& 0\\ 0& 2 & 0& -1& 0& 0& 0\\ -1& 0 & 2& -1& 0& 0& 0\\ 0& -1 & -1& 2& -1& 0& 0\\0& 0& 0 & -1& 2& -1& 0\\ 0& 0& 0& 0 & -1& 2& -1\\ 0&0& 0& 0& 0 & -1& 2\end{pmatrix}\]

23.6 The root system \(E_6\)↩︎

Note that these roots (unlike \(\alpha_8={\boldsymbol{e}}_2-{\boldsymbol{e}}_3\) and \(\alpha_7={\boldsymbol{e}}_3-{\boldsymbol{e}}_4\)) satisfy the equations \(x_1+x_2=0, x_2-x_3=0\). Thus \(E_6\) is the intersection of \(E_8\) with this subspace. So it includes the roots \(\pm {\boldsymbol{e}}_i\pm {\boldsymbol{e}}_j\) with \(4\le i,j\le 8\) distinct (40 roots), and \(\sum_{i=1}^8 (\pm \frac{1}{2}{\boldsymbol{e}}_i)\) with even number of minuses and the opposite signs for \({\boldsymbol{e}}_1\) and \({\boldsymbol{e}}_2\) and for \({\boldsymbol{e}}_1\) and \({\boldsymbol{e}}_3\) (32 roots). Altogether we get 72 roots. The Cartan matrix is the upper left corner 6 by 6 submatrix of the Cartan matrix of \(E_8\), so it is \[A=\begin{pmatrix} 2& 0 & -1& 0& 0& 0\\ 0& 2 & 0& -1& 0& 0\\ -1& 0 & 2& -1& 0& 0\\ 0& -1 & -1& 2& -1& 0\\0& 0& 0 & -1& 2& -1\\ 0& 0& 0& 0 & -1& 2\end{pmatrix}\] This recovers the Dynkin diagram \(E_6\).

23.7 The elements \(\rho\) and \(\rho^\vee\)↩︎

Recall that the elements \(\rho\in \mathfrak{h}^*\) and \(\rho^\vee\in \mathfrak{h}\) for a simple Lie algebra \(\mathfrak{g}\) are defined by the conditions \((\rho,\alpha_i^\vee)=(\rho^\vee,\alpha_i)=1\) for all \(i\) (note that \(\rho\) is not a root in general, and \(\rho^\vee\) is not an instance of the assignment \(\alpha\mapsto \alpha^\vee\) for roots \(\alpha\)). So for classical Lie algebras they can be computed from Example 55. Namely, we get \[\rho_{A_{n-1}}=\rho_{A_{n-1}}^\vee=(\tfrac{n-1}{2},\tfrac{n-3}{2},...,-\tfrac{n-1}{2}),\] \[\rho_{B_n}=\rho_{C_n}^\vee=(\tfrac{2n-1}{2},...,\tfrac{3}{2},\tfrac{1}{2}),\] \[\rho_{C_n}=\rho_{B_n}^\vee=(n,n-1,...,1),\] \[\rho_{D_n}=\rho_{D_n}^\vee=(n-1,n-2,...,0).\]

23.8 Proof of Theorem 149↩︎

Now that we have shown that there exist root systems attached to all Cartan matrices, it remains to classify Cartan matrices (or Dynkin diagrams), i.e. show that there are no others than those we have considered. For this purpose we consider Dynkin diagrams as graphs with certain kind of special edges (with one, two or three lines and a possible orientation). Note first that any subgraph of a Dynkin diagram must itself be a Dynkin diagram, since a principal submatrix of a positive definite symmetric matrix is itself positive definite. On the other hand, consider untwisted and twisted affine Dynkin diagrams depicted on the first picture at https://en.wikipedia.org/wiki/Affine_Lie_algebra. These are not Dynkin diagrams since the corresponding matrix \(A\) is degenerate, hence not positive definite.

Thus they cannot occur inside a Dynkin diagram (as the restriction of a positive definite inner product to a subspace must be positive definite).

We conclude that a Dynkin diagram is a tree. Indeed, it cannot have a loop with simple edges, since this is the affine diagram \(\widetilde{A}_{n-1}\), which has a null vector \((1,...,1)\). If there is a loop with non-simple edges, this is even worse - this vector will have a negative inner product with itself.

Further, it cannot have vertices with more than four simple edges coming out since it cannot have a subdiagram \(\widetilde{D}_4\) (and for non-simple edges it is even worse, as before). Thus all the vertices of our tree are i-valent for \(i\le 3\).

Also we cannot have a subdiagram \(\widetilde{D}_n\), \(n\ge 5\), which implies that there is at most one trivalent vertex.

Further, if there is a triple edge then the diagram is \(G_2\). There is no way to attach any edge to the \(G_2\) diagram because \(D_4^{(3)}\) and \(\widetilde{G}_2\) are forbidden.

Next, if there is a trivalent vertex then there cannot be a non-simple edge anywhere in the diagram (as we have forbidden affine diagrams \(A_{2k-1}^{(2)},\widetilde{B}_n\)). So in this case the diagram is simply laced, so it must be on our list (\(D_n,E_6,E_7,E_8\)) since it cannot contain affine diagrams \(\widetilde{E}_6,\widetilde{E}_7,\widetilde{E}_8\).

It remains to consider chain-shaped diagrams. They can’t contain two double edges (affine diagrams \(A_{2k}^{(2)},D_{k+1}^{(2)},\widetilde{C}_n\)). Thus if the double edge is at the end, we can only get \(B_n\) and \(C_n\).

Finally, if the double edge is in the middle, we can’t have affine subdiagram \(\widetilde{F}_4\) or \(E_6^{(2)}\), so our diagram must be \(F_4\). Theorem 149 is proved.

23.9 Simply laced and non-simply laced diagrams↩︎

As we already mentioned, a Dynkin diagram (or the corresponding root system) is called simply laced if all the edges are simple, i.e. \(a_{ij}=0,-1\) for \(i\ne j\). This is equivalent to the Cartan matrix being symmetric, or to all roots having the same length. The connected simply-laced diagrams are \(A_n,n\ge 1; D_n, n\ge 4; E_6,E_7,E_8\). The remaining diagrams \(B_n,C_n,F_4,G_2\) are not simply laced, but they contain roots of only two squared lengths, whose ratio is \(2\) for double edge (\(B_n,C_n,F_4\)) and \(3\) for triple edge (\(G_2\)). The roots of the bigger length are called long and of the smaller length are called short.

It is easy to see that long and short roots form a root system of the same rank (but not necessarily irreducible). For instance, in \(G_2\) both form a root system of type \(A_2\), and in \(B_2\) both are \(A_1\times A_1\). In \(B_3\) long roots form \(D_3\) and short ones form \(A_1\times A_1\times A_1\). However, only long roots form a root subsystem, since a long positive root can be the sum of two short ones, but not vice versa.

24 Construction of a semisimple Lie algebra from a Dynkin diagram↩︎

24.1 Serre relations↩︎

Let \(\boldsymbol{k}\) be an algebraically closed field of characteristic zero. We would like to show that any reduced root system gives rise to a semisimple Lie algebra over \(\boldsymbol{k}\), and moreover a unique one. To this end, it suffices to show that any reduced irreducible root system gives rise to a unique (finite dimensional) simple Lie algebra.

Let \(\mathfrak{g}\) be a finite dimensional simple Lie algebra over \({\boldsymbol{k}}\) with Cartan subalgebra \(\mathfrak{h}\subset \mathfrak{g}\) and root system \(R\subset \mathfrak{h}^*\) (which is thus reduced and irreducible). Fix a polarization of \(R\) with the set of simple roots \(\Pi=(\alpha_1,...,\alpha_r)\), and let \(A=(a_{ij})\) be the Cartan matrix of \(R\). We have a decomposition \(\mathfrak{g}= \mathfrak{n}_+\oplus \mathfrak{h}\oplus \mathfrak{n}_-\), where \(\mathfrak{n}_\pm:=\oplus_{\alpha\in R_{\pm}}\mathfrak{g}_\alpha\) are the Lie subalgebras spanned by positive, respectively negative root vectors. Pick elements \(e_i\in \mathfrak{g}_{\alpha_i}\), \(f_i\in \mathfrak{g}_{-\alpha_i}\) so that \(e_i,f_i,h_i=[e_i,f_i]\) form an \({\mathfrak{sl}}_2\)-triple.

The last two sets of relations are called Serre relations. Note that if \(a_{ij}=0\) then the Serre relations just say that \([e_i,e_j]=[f_i,f_j]=0\).

24.2 The Serre presentation for semisimple Lie algebras↩︎

Now for any reduced root system \(R\) let \(\mathfrak{g}(R)\) be the Lie algebra generated by \(e_i,f_i,h_i,i=1,...,r\), with defining relations being the relations of Theorem 157. Precisely, this means that \(\mathfrak{g}(R)\) is the quotient of the free Lie algebra \(FL_{3r}\) with generators \(e_i,f_i,h_i\) modulo the Lie ideal generated by the differences of the left and right hand sides of these relations.

25 Representation theory of semisimple Lie algebras↩︎

25.1 Representations of semisimple Lie algebras↩︎

We will now develop representation theory of complex semisimple Lie algebras. The representation theory of semisimple Lie algebras over an algebraically closed field of characteristic zero is completely parallel, so we will stick to the complex case. So all representations will be over \(\Bbb C\). We will mostly be interested in finite dimensional representations; as we know, they can be exponentiated to holomorphic representations of the corresponding simply connected Lie group \(G\), which defines a bijection between isomorphism classes of such representations of \(\mathfrak{g}\) and \(G\).

Let \(\mathfrak{g}\) be a semisimple Lie algebra. Recall that by Theorem 122, every finite dimensional representation of \(\mathfrak{g}\) is completely reducible, so to classify finite dimensional representations it suffices to classify irreducible representations.

As in the simplest case of \(\mathfrak{sl}_2\), a crucial tool is the decomposition of a representation in a direct sum of eigenspaces of a Cartan subalgebra \(\mathfrak{h}\subset \mathfrak{g}\).

It is easy to see that \(\mathfrak{g}_\alpha V[\lambda]\subset V[\lambda+\alpha]\).

Let \(V'\subset V\) be the span of all weight vectors in \(V\). Then it is clear that \(V'=\oplus_{\lambda\in \mathfrak{h}^*}V[\lambda]\).

Note that not every representation of \(\mathfrak{g}\) has a weight decomposition (e.g., for \(V=U(\mathfrak{g})\) with \(\mathfrak{g}\) acting by left multiplication all weight subspaces are zero).

25.2 Verma modules↩︎

Even though we are mostly interested in finite dimensional representations of \(\mathfrak{g}\), it is useful to consider some infinite dimensional representations, which are called Verma modules.

The Verma module \(M_\lambda\) is defined as “the largest highest weight representation with highest weight \(\lambda\)". Namely, it is generated by a single highest weight vector \(v_\lambda\) with defining relations \(hv=\lambda(h)v\) for \(h\in \mathfrak{h}\) and \(e_iv=0\). More formally speaking, we make the following definition.

In this realization, the highest weight vector \(v_\lambda\) is just the class of the unit \(1\) of \(U(\mathfrak{g})\).

Recall that \(Q_+\) denotes the set of elements \(\sum_{i=1}^r k_i\alpha_i\) where \(k_i\in \Bbb Z_{\ge 0}\). We obtain

25.3 Finite dimensional modules↩︎

Since every finite dimensional irreducible \(\mathfrak{g}\)-module is highest weight, it is of the form \(L_\lambda\) for \(\lambda\) belonging to some subset \(P_F\subset P\), the set of weights \(\lambda\) such that \(L_\lambda\) is finite dimensional. So to obtain a final classification of finite dimensional irreducible representations of \(\mathfrak{g}\), we should determine the subset \(P_F\).

Let \(P_+\subset P\) be the intersection of \(P\) with the closure of the dominant Weyl chamber \(C_+\); i.e., \(P_+\) is the set of nonnegative integer linear combinations of the fundamental weights \(\omega_i\). In other words, \(P_+\) is the set of \(\lambda\in P\) such that \((\lambda,\alpha_i^\vee)\in \Bbb Z_{\ge 0}\) for \(1\le i\le r\). Weights belonging to \(P_+\) are called dominant integral.

Now we are ready to state the main classification theorem.

26 The Weyl character formula↩︎

26.1 Characters↩︎

Let \(V\) be a finite dimensional representation of a semisimple Lie algebra \(\mathfrak{g}\). Recall that the action of \(\mathfrak{g}\) on \(V\) can be exponentiated to the action of the corresponding simply connected complex Lie group \(G\). Recall also that the character of a finite dimensional representation \(V\) of any group \(G\) is the function \[\chi_V(g)={\rm Tr}|_V(g).\]

Let us compute this character in our case. To this end, let \(\mathfrak{h}\subset \mathfrak{g}\) be a Cartan subalgebra, \(h\in \mathfrak{h}\), and let us compute \(\chi_V(e^h)\). Note that this completely determines \(\chi_V\) since it determines \(\chi_V(e^x)\) for any semisimple element \(x\in \mathfrak{g}\) (as such an element can be conjugated into \(\mathfrak{h}\)), and semisimple elements form a dense open set in \(\mathfrak{g}\) (complement of zeros of some polynomial). So elements of the form \(e^x\) as above form a dense open set at least in some neighborhood of \(1\) in \(G\), and an analytic function on \(G\) is determined by its values on any nonempty open set.

We know that \(V\) has a weight decomposition: \(V=\oplus_{\mu\in P}V[\mu]\). Thus we have \[\chi_V(e^h)=\sum_{\mu\in P}\dim V[\mu] e^{\mu(h)}.\] Consider the group algebra \(\Bbb Z[P]\). It sits naturally inside the algebra of analytic functions on \(\mathfrak{h}\) via \(\lambda\mapsto e^\lambda\), where \(e^\lambda(h):=e^{\lambda(h)}\), and we see that \(\chi_V\in \Bbb Z [P]\), namely \[\chi_V=\sum_{\mu\in P}\dim V[\mu] e^\mu.\] We will call the element \(\chi_V\) the character of \(V\).

26.2 Category \(\mathcal{O}\)↩︎

Note that the above definition of character is a purely formal algebraic definition, i.e., \(\chi_V\) is simply the generating function of dimensions of weight subspaces of \(V\). So it makes sense for any (possibly infinite dimensional) representation \(V\) with a weight decomposition into finite dimensional weight subspaces, except we may obtain an infinite sum. More precisely, we make the following definition.

Here the subscript “int" indicates that we consider only integral weights (i.e., ones in \(P\)). However, for brevity we will drop this subscript in this section and just denote this category by \(\mathcal{O}\).

For example, any highest weight module with integral highest weight belongs to \(\mathcal{O}\).

Let \(\mathcal{R}\) be the ring of series \(a:=\sum_{\mu\in P} a_\mu e^\mu\) (\(a_\mu\in \Bbb Z\)) such that the set \(P(a)\) of \(\mu\) with \(a_\mu\ne 0\) is contained in the union of sets \(\lambda^i-Q_+\) for a finite collection of weights \(\lambda^1,...,\lambda^N\in P\). Then for every \(V\in \mathcal{O}\) we can define the character \(\chi_V\in \mathcal{R}\). Moreover, it is easy to see that if \[0\to X\to Y\to Z\to 0\] is a short exact sequence in \(\mathcal{O}\) then \(\chi_Y=\chi_X+\chi_Z\), and that for any \(V,U\in \mathcal{O}\) we have \(V\otimes U\in \mathcal{O}\) and \(\chi_{V\otimes U}=\chi_V\chi_U\).

Note that we have a homomorphism \(\varepsilon: W\to \Bbb Z/2\) given by the formula \(w\mapsto \det(w|_\mathfrak{h})\), i.e. \(w\mapsto (-1)^{\ell(w)}\); it is defined on simple reflections by \(s_i\mapsto -1\). This homomorphism is called the sign character. For example, for type \(A_{n-1}\) this is the sign of a permutation in \(S_n\). We will say that an element \(f\in \Bbb C[P]\) is anti-invariant under \(W\) if \(w(f)=(-1)^{\ell(w)}f\) for all \(w\in W\).

26.3 The Weyl character formula↩︎

The proof of this theorem is in the next subsection.

For example, for \(\mathfrak{g}=\mathfrak{sl}_n\) Corollary 46 reduces to the usual product formula for the Vandermonde determinant.

26.4 Proof of the Weyl character formula↩︎

Consider the product \(\Delta \chi_\lambda\in \Bbb Z[P]\). We know that \(\chi_\lambda\) is \(W\)-invariant, so this product is \(W\)-anti-invariant. Thus, \[\Delta \chi_\lambda=\sum_{\mu\in P} c_\mu e^\mu,\] where \(c_{w\mu}=(-1)^{\ell(w)}c_\mu\). Moreover, \(c_\mu=0\) unless \(\mu\in \lambda+\rho-Q_+\), and \(c_{\lambda+\rho}=1\). Thus to prove the Weyl character formula, we need to show that \(c_\mu=0\) if \(\mu\in P_+\cap (\lambda+\rho-Q_+)\) and \(\mu\ne \lambda+\rho\).

To this end, we will construct the above decomposition \(\Delta \chi_\lambda\) using representation theory, so that this vanishing property is apparent from the construction.

First recall from Subsection 18.3 that we have the Casimir element \(C\) of \(U(\mathfrak{g})\) given by the formula \(C=\sum_i a_ia^i\) for a basis \(a_i\in \mathfrak{g}\) with dual basis \(a^i\) of \(\mathfrak{g}\) under the Killing form. This element is central, so acts by a scalar on every highest weight (in particular, finite dimensional irreducible) representation. We can write \(C\) in the form \[C=\sum_j x_j^2+\sum_{\alpha\in R_+}(e_{-\alpha}e_\alpha+e_\alpha e_{-\alpha}),\] for an orthonormal basis \(x_j\) of \(\mathfrak{h}\) and \(e_\alpha\) are normalized in such a way that \([e_\alpha,e_{-\alpha}]=h_\alpha\). Hence \[C=\sum_j x_j^2+2\sum_{\alpha\in R_+}e_{-\alpha}e_\alpha+\sum_{\alpha\in R_+}h_\alpha.\] Thus we get

Now we will define a sequence of modules \(K(b)\) from category \(\mathcal{O}\) parametrized by some binary strings \(b\). This is done inductively. We set \(K(\emptyset)=L_\lambda\). Now suppose \(K(b)\) is already defined. If \(K(b)=0\) then we set \(K(b0)=K(b1)=0\). Otherwise, pick a nonzero vector \(v_b\in K(b)\), of some weight \(\nu(b)\in \lambda-Q_+\) such that the height of \(\lambda-\nu(b)\) takes the minimal possible value. Then \(v_b\) is a highest weight vector, and we can consider the corresponding homomorphism \[\xi_b: M_{\nu_b}\to K(b).\] Let \(K(b1),K(b0)\) be the kernel and cokernel of \(\xi_b\). We have \[\chi_{K(b1)}-\chi_{M_{\nu(b)}}+\chi_{K(b)}-\chi_{K(b0)}=0.\] Thus we have \[\chi_{K(b)}=\chi_{M_{\nu(b)}}-\chi_{K(b1)}+\chi_{K(b0)}.\]

Now, it is clear that for every \(\mu\), every sufficiently long sequence \(b\) satisfies \(K(b)[\mu]=0\). So iterating this formula starting with \(b=\emptyset\), we will get \[\label{seri} \chi_\lambda=\sum_b (-1)^{\Sigma(b)}\chi_{M_{\nu(b)}}\tag{9}\] where \(\Sigma(b)\) is the sum of digits of \(b\) (which could a priori be an infinite sum). So \[\Delta \chi_\lambda=\sum_b (-1)^{\Sigma(b)}e^{\nu(b)+\rho}.\] Also note that by induction in the length of \(b\) we can conclude that the eigenvalue of \(C\) on \(M_{\nu(b)}\) is \(|\lambda+\rho|^2-|\rho|^2\) regardless of \(b\), which implies that \[|\nu(b)+\rho|^2=|\lambda+\rho|^2\] for all \(b\); in particular, this shows that the sum 9 is finite.

So it remains to show that if \(\mu=\lambda+\rho-\beta \in P_+\) with \(\beta\in Q_+\) and \(\beta\ne 0\) then \(|\mu|^2<|\lambda+\rho|^2\). Indeed, \[|\lambda+\rho|^2-|\mu|^2=|\lambda+\rho|^2-|\lambda-\beta+\rho|^2=\] \[2(\lambda+\rho,\beta)-|\beta|^2>(\lambda+\rho,\beta)-|\beta|^2=(\lambda+\rho-\beta,\beta)\ge 0.\] This completes the proof of the Weyl character formula.

26.5 The Weyl dimension formula↩︎

Recall that the Weyl character formula can be written as a trace formula: for \(h\in \mathfrak{h}\) \[\chi_{\lambda}(e^h)=\text{Tr}|_{L_\lambda}(e^h)=\frac{\sum_{w\in W}(-1)^{\ell(w)}e^{(w(\lambda+\rho),h)}}{\prod_{\alpha\in R_+}(e^{\frac{1}{2}(\alpha,h)}-e^{-\frac{1}{2}(\alpha,h)})}.\] The dimension of \(L_\lambda\) should be obtained from this formula when \(h=0\). However, we do not immediately get the answer since this formula gives the character as a ratio of two trigonometric polynomials which both vanish at \(h=0\), giving an indeterminacy. We know the limit exists since the character is a trigonometric polynomial, but we need to compute it. This can be done as follows.

Let us restrict attention to \(h=2th_\rho\) where \(t\in \Bbb R\) and \(h_\rho\in \mathfrak{h}\) corresponds to \(\rho\in \mathfrak{h}^*\) using the identification induced by the invariant form. We have \[\chi_{\lambda}(e^{2th_\rho})=\frac{\sum_{w\in W}(-1)^{\ell(w)}e^{2t(w(\lambda+\rho),\rho)}}{\prod_{\alpha\in R_+}(e^{t(\alpha,\rho)}-e^{-t(\alpha,\rho)})}.\] The key idea is that for this specialization the numerator can also be factored using the denominator formula, which will allow us to resolve the indeterminacy. Namely, we have \[\label{princ} \chi_{L_\lambda}(e^{2th_\rho})=\frac{\prod_{\alpha\in R_+}(e^{t(\alpha,\lambda+\rho)}-e^{-t(\alpha,\lambda+\rho)})}{\prod_{\alpha\in R_+}(e^{t(\alpha,\rho)}-e^{-t(\alpha,\rho)})}.\tag{10}\] Now sending \(t\to 0\), we obtain

Note that this number is an integer, but this is not obvious without its interpretation as the dimension of a representation.

Formula 10 has a meaning even before taking the limit. Namely, the eigenvalues of the element \(2h_\rho\) define a \(\Bbb Z\)-grading on the representation \(L_\lambda\) called the principal grading, and we obtain a product formula for the Poincaré polynomial of this grading.

Lie groups and Lie algebras II

27 Representations of \(GL_n\), I↩︎

We begin with a more detailed study of finite dimensional representations of semisimple Lie algebras and the corresponding complex Lie groups.

27.1 Tensor products of fundamental representations↩︎

The following result shows that if we understand fundamental representations of a semisimple Lie algebra \(\mathfrak{g}\) (i.e., irreducible representations with fundamental highest weights \(\omega_i\)), we can gain some insight into general finite dimensional representations.

27.2 Representations of \(SL_n(\Bbb C)\)↩︎

Let us now discuss more explicitly the representation theory of \(SL_n(\Bbb C)\). We will consider its finite dimensional complex analytic representations as a complex Lie group. We have shown that this is equivalent to considering finite dimensional representations of the Lie algebra \(\mathfrak{sl}_n(\Bbb C)\). We have also seen that these are completely reducible and the irreducible representations are \(L_\lambda\), where \(\lambda=\sum_{i=1}^{n-1}m_i\omega_i\), \(\omega_i\) are the fundamental weights, and \(m_i\in \Bbb Z_{\ge 0}\).

First let us compute \(\omega_i\). Recall that the standard Cartan subalgebra \(\mathfrak{h}\) is the space \(\Bbb C_0^n\) of vectors in \(\Bbb C^n\) with zero sum of coordinates (diagonal matrices with trace zero). So elements of \(\mathfrak{h}^*\) can be viewed as vectors \((x_1,...,x_n)\in \Bbb C^n\) modulo simultaneous shift of all coordinates by the same number (i.e., \(\mathfrak{h}^*=\Bbb C^n/\Bbb C_{\rm diag}\)).

Recall that the simple coroots are \(\alpha_i^\vee=\boldsymbol{e}_i-\boldsymbol{e}_{i+1}\). Thus \(\omega_i\) are determined by the conditions \[(\omega_i,\boldsymbol{e}_j-\boldsymbol{e}_{j+1})=\delta_{ij}.\] This means that \(\omega_i=(1,...,1,0,...,0)\) where there are \(i\) copies of \(1\). Thus a dominant integral weight \(\lambda\) has the form \[\lambda=(m_1+...+m_{n-1},m_2+...+m_{n-1},....,m_{n-1},0).\] So dominant integral weights are parametrized by non-increasing sequences \(\lambda_1\ge...\ge \lambda_{n-1}\) of nonnegative integers. This agrees with the representation theory of \(SL_2(\Bbb C)\) that we worked out before: in this case the sequence has just one term.

Let us now describe explicitly the fundamental representations \(L_{\omega_i}\). Consider first the representation \(V=\Bbb C^n\) with the usual action of matrices. It is called the vector representation or the tautological representation (as every matrix goes to itself). It is irreducible and has a standard basis \(v_1,...,v_n\). To find its highest weight, we have to find a vector \(v\ne 0\) such that \(e_iv=0\). As \(e_i=E_{i,i+1}\), we have \(v=v_1\). It is easy to see that \(hv=\omega_1(h)v\), so we see that \(v\) has weight \(\omega_1\), hence \(L_{\omega_1}=V\).

To construct \(L_{\omega_m}\) for \(m>1\), consider the exterior power \(\wedge^m V\). It is easy to show that it is irreducible. A basis of \(\wedge^m V\) consists of wedges \(v_{i_1}\wedge...\wedge v_{i_m}\) where \(i_1<...<i_m\). The highest weight vector is clearly \(v_1\wedge...\wedge v_m\), and it has weight \(\omega_m\). Thus \(L_{\omega_m}=\wedge^m V\).

Note that \(\wedge^n V=\Bbb C\) (the trivial representation) since every matrix in \(SL_n(\Bbb C)\) acts by its determinant, which is \(1\), and \(\wedge^mV=0\) for \(m>n\). Also \(V^*\cong \wedge^{n-1}V\) since the wedge pairing \(V\otimes \wedge^{n-1}V\to \wedge^nV=\Bbb C\) is invariant and nondegenerate. Similarly, \(\wedge^mV^*\cong \wedge^{n-m}V\).

We now see from Proposition 171 that the irreducible representation \(L_\lambda\) for \(\lambda=\sum_i m_i\omega_i\) is generated inside \(\otimes_{i=1}^{n-1} (\wedge^i V)^{\otimes m_i}\) by the tensor product of the highest weight vectors.

27.3 Representations of \(GL_n(\Bbb C)\)↩︎

Let us now explain how to extend these results to \(GL_n(\Bbb C)\). This is easy to do since \(GL_n(\Bbb C)\) is not very different from the direct product \(\Bbb C^\times\times SL_n(\Bbb C)\). Namely, \(GL_n(\Bbb C)=(\Bbb C^\times\times SL_n(\Bbb C))/\mu_n\) where \(\mu_n\) is the group of roots of unity of order \(n\) embedded as \(z\mapsto (z^{-1},z\boldsymbol{1}_n)\). Indeed, the corresponding covering homomorphism \(\Bbb C^\times\times SL_n(\Bbb C)\to GL_n(\Bbb C)\) is given by \((z,A)\mapsto zA\). So it suffices to classify irreducible holomorphic representations of the complex Lie group \(\Bbb C^\times\times SL_n(\Bbb C)\); the irreducible holomorphic representations of \(GL_n(\Bbb C)\) are a subset of them.

For \(n=1\) this is just the problem of describing the holomorphic representations of \(\Bbb C^\times\). This is easy. The Lie algebra is spanned by a single element \(h\) such that \(e^{2\pi ih}=1\). This element must act in a representation by an operator \(H\) such that \(e^{2\pi iH}=1\). It follows that \(H\) is diagonalizable with integer eigenvalues. Thus representations of \(\Bbb C^\times\) are completely reducible, with irreducibles \(\chi_N\) one-dimensional and labeled by integers \(N\in \Bbb Z\), \(\chi_N(z)=z^N\).

The same argument leads to a similar answer for \(\Bbb C^\times \times SL_n\): representations are completely reducible with irreducibles being \(L_{\lambda,N}=\chi_N\otimes L_\lambda\). Moreover, the ones factoring through \(GL_n\) just have \(N=nr+\sum_{i=1}^{n-1} \lambda_i\) for some integer \(r\).

Recall that \(GL_n\) has reductive Lie algebra \(\mathfrak{gl}_n\) with Cartan subalgebra \(\mathfrak{h}=\Bbb C^n\). The highest weight of \(L_{\lambda,nm_n+\sum_{i=1}^{n-1} \lambda_i}\) is easily computed and equals \((m_1+...+m_{n-1}+m_n,...,m_{n-1}+m_n,m_n)\). Thus highest weights of finite dimensional representations are non-increasing sequences \((\lambda_1,...,\lambda_n)\) of integers which don’t have to be positive. The fundamental representations are still \(L_{\omega_m}=\wedge^m V\), and the only difference with \(SL_n\) is that now the top exterior power \(\wedge^n V\) is not trivial but rather is a 1-dimensional determinant character with highest weight \(\omega_n=(1,...,1)\). The highest weight of a finite dimensional representation then has the form \(\lambda=\sum_{i=1}^n m_i\omega_i\), where \(m_i\ge 0\) for \(i\ne n\), while \(m_n\) is an arbitrary integer. Consequently, \(L_\lambda\) is found inside \(\otimes_{i=1}^n (\wedge^i V)^{\otimes m_i}\) as the representation generated by the product of highest weight vectors. Note that it makes sense to take \(m_n<0\), as for a one-dimensional representation and \(k<0\) it is natural to define \(\chi^{\otimes k}:=(\chi^*)^{\otimes -k}\).

The representations with \(m_n\ge 0\) are especially important; it is easy to see that these are exactly the ones that occur inside \(V^{\otimes N}\) for some \(N\) (check it!). These representations are called polynomial since their matrix coefficients are polynomial functions of the matrix entries \(x_{ij}\) of \(X\in GL_n(\Bbb C)\), and consequently they extend by continuity to representations of the semigroup \({\rm Mat}_n(\Bbb C)\supset GL_n(\Bbb C)\). Note that any irreducible representation is a polynomial one tensored with a non-positive power of the determinant character \(\wedge^nV\).

27.4 Schur-Weyl duality↩︎

Note that highest weights of polynomial representations are non-increasing sequences of nonnegative integers \((\lambda_1,...,\lambda_n)\), i.e. partitions with \(\le n\) parts. Namely, they are partitions of \(|\lambda|=\sum_i \lambda_i\), which is just the eigenvalue of \(\boldsymbol{1}_n\in {\mathfrak{gl}}_n\) on \(L_\lambda\) and can also be defined as the number \(N\) such that \(L_\lambda\) occurs in \(V^{\otimes N}\).

Traditionally partitions are encoded by Young diagrams. Namely, the Young diagram of a partition \(\lambda=(\lambda_1,...,\lambda_n)\) consists of \(n\) rows of boxes, the \(i\)-th row consisting of \(\lambda_i\) boxes, so that row \(i\) is placed directly under row \(i-1\) and all rows start on the same vertical line. For example, here are the Young diagrams of the partitions \((4,3,2)\) (left) and \((3,3,2,1)\) (right):

Thus we have \[V^{\otimes N}=\oplus_{\lambda: |\lambda|=N} L_\lambda\otimes \pi_\lambda,\] where \(\pi_\lambda:=\mathrm{Hom}_{GL_n(\Bbb C)}(L_\lambda,V^{\otimes N})\) are multiplicity spaces. Here the summation is over partitions of \(N\), and \(L_\lambda=0\) if \(\lambda\) has more than \(n\) parts. To understand the spaces \(\pi_\lambda\), note that the symmetric group \(S_N\) acts on \(V^{\otimes N}\) and commutes with \(GL_n(\Bbb C)\), so it gets to act on each \(\pi_\lambda\).

Let \(A\) be the image of \(U(\mathfrak{gl}_n)\) in \({\rm End}_{\Bbb C}(V^{\otimes N})\), and \(B\) be the image there of \(\Bbb CS_N\). The algebras \(A,B\) commute.

Thus we see that representations of \(S_N\) are labeled by partitions \(\lambda\) of \(N\), and those that occur in \(V^{\otimes N}\) correspond to the partitions that have \(\le \dim V\) parts. Moreover, we claim that this labeling of representations by partitions does not depend on \(\dim V\). To show this, suppose \(\lambda\) has \(\le n\) parts and \(V=\Bbb C^n\). We have the Schur-Weyl decomposition of \(GL_{n+1}(\Bbb C)\times S_N\)-modules \[(V\oplus \Bbb C)^{\otimes N}=\oplus_\mu L_\mu^{(n+1)}\otimes \pi_\mu^{(n+1)},\] Let us restrict this sum to \(GL_n(\Bbb C)\times S_N\), and consider what happens to the summand \(L_\lambda^{(n+1)}\otimes \pi_\lambda^{(n+1)}\). The highest weight vector \(v\) in \(L_\lambda^{(n+1)}\) tensored with any element \(w\) of \(\pi_\lambda^{(n+1)}\) sits in \(V^{\otimes N}\subset (V\oplus \Bbb C)^{\otimes N}\), since the \(n+1\)-th component of its weight is zero. Hence \(v\otimes w\) generates a copy of \(L_\lambda^{(n)}\otimes \pi_\lambda^{(n)}\) as a \(GL_n(\Bbb C)\times S_N\)-module. This implies that \(\pi_\lambda^{(n+1)}\cong \pi_\lambda^{(n)}\).

28 Representations of \(GL_n\), II↩︎

28.1 Schur functors↩︎

Thus we have \[V^{\otimes N}=\oplus_{\lambda}S^\lambda V\otimes \pi_\lambda,\] and if \(\lambda\) has \(\le n\) parts and \(V=\Bbb C^n\) then \(S^\lambda V=L_\lambda\) as a representation of \(GL(V)=GL_n(\Bbb C)\).

Let us compute the dimension of \(S^\lambda V\) when \(\dim V=N\) and \(\lambda\) has \(k\) parts. We have \(\rho=(N-1,N-2,...,1,0)\) (for \(SL_N\)), so the Weyl dimension formula tells us that \[\dim S^\lambda V=\prod_{1\le i<j\le N}\frac{\lambda_i-\lambda_j+j-i}{j-i}=\] \[\prod_{1\le i<j\le k}\frac{\lambda_i-\lambda_j+j-i}{j-i}\prod_{1\le i\le k<j\le N}\frac{\lambda_i+j-i}{j-i}=\] \[\prod_{1\le i<j\le k}\frac{\lambda_i-\lambda_j+j-i}{j-i}\prod_{i=1}^k \frac{(N+1-i)...(N+\lambda_i-i)}{(k+1-i)...(k+\lambda_i-i)}.\] We obtain

Moreover, we see that \(P_\lambda(N)\) is an integer-valued polynomial, i.e., it takes integer values at integer points (this is equivalent to being an integer linear combination of \(\binom{N}{j}\)).

28.2 The fundamental theorem of invariant theory↩︎

Suppose we have a finite dimensional vector space \(V\) and a collection of tensors \(T_i\in V^{\otimes m_i}\otimes V^{*\otimes n_i}\), \(i=1,...,k\). An important problem is to describe “coordinate free" invariants of such a collection of tensors, i.e., polynomial functions \(F(T_1,...,T_k)\) which are invariant under the action of \(GL(V)\). How can we classify such functions? This sounds formidably hard in such generality, but turns out to be very easy using Schur-Weyl duality.

It suffices to study such functions that have homogeneity degree \(d_i\) with respect to each \(T_i\). To do so, we will depict each \(T_i\) by a vertex with \(m_i\) incoming and \(n_i\) outgoing arrows. We should think of incoming arrows as \(V\)-components and outgoing ones as \(V^*\)-components. Let us draw \(d_i\) such vertices for each \(i\). To construct an invariant, let us connect the arrows preserving orientation so that all the arrows are used (this will only be possible if the number of incoming arrows equals the number of outgoing ones; otherwise every invariant of the multidegree \((d_1,...,d_k)\) will be zero). To the obtained graph \(\Gamma\) we can assign the convolution of tensors, which gives an invariant function \(F_\Gamma\) of the correct multidegree.

29 Representations of \(GL_n\), III↩︎

29.1 Schur polynomials and characters of representations of the symmetric group↩︎

Using Schur-Weyl duality and the character formula for representations of \(GL_n\), we can obtain information about characters of the symmetric group. Namely, it follows from the Weyl character formula that the characters of representations of \(GL_n\) are given by the formula \[s_\lambda(x_1,...,x_n)=\frac{\sum_{\sigma\in S_n}{\rm sign}(\sigma)x_{\sigma(1)}^{\lambda_1+n-1}...x_{\sigma(n)}^{\lambda_n}}{\prod_{i<j}(x_i-x_j)}=\frac{\det(x_i^{\lambda_j+n-j})}{\prod_{i<j}(x_i-x_j)}.\] These symmetric polynomials are called Schur polynomials. For example, the character of \(S^mV\) is \[s_{(m)}(x_1,...,x_n)=\sum_{1\le j_1\le ...\le j_m\le n}x_{j_1}...x_{j_m}=h_m(x_1,...,x_n),\] the \(m\)-th complete symmetric function, and the character of \(\wedge^mV\) is \[s_{(1^m)}(x_1,...,x_n)=\sum_{1\le j_1<...<j_m\le n}x_{j_1}...x_{j_m}=e_m(x_1,...,x_n),\] the \(m\)-th elementary symmetric function.

Let us now compute the trace in \(V^{\otimes N}\) of \(x^{\otimes N}\sigma\), where \(x={\rm diag}(x_1,...,x_n)\) is a diagonal matrix and \(\sigma\in S_N\) a permutation. Let \(\sigma\) have \(m_i\) cycles of length \(i\). Then we have \[{\rm Tr}|_{V^{\otimes N}}(x^{\otimes N}\sigma)=\prod_i (x_1^i+...+x_n^i)^{m_i}.\] On the other hand, using Schur-Weyl duality, we get \[{\rm Tr}|_{V^{\otimes N}}(x^{\otimes N}\sigma)=\sum_\lambda \chi_\lambda(\sigma)s_\lambda(x),\] where \(\chi_\lambda(\sigma)={\rm Tr}|_{\pi_\lambda}(\sigma)\) is the character of the representation \(\pi_\lambda\) of \(S_N\). Thus we have \[\sum_\lambda \chi_\lambda(\sigma)s_\lambda(x)=\prod_i (x_1^i+...+x_n^i)^{m_i}.\] Multiplying this by the discriminant, we get \[\sum_\lambda \chi_\lambda(\sigma)\det(x_i^{\lambda_j+n-j})=\prod_{i<j}(x_i-x_j)\cdot \prod_i (x_1^i+...+x_n^i)^{m_i}.\] Thus we get

29.2 Howe duality↩︎

Howe duality is another instance when we have a double centralizer property. Consider two finite dimensional complex vector spaces \(V,W\), and consider the symmetric power \(S^N(V\otimes W)\) as a representation of \(GL(V)\times GL(W)\).

Note that if \(\lambda\) has more parts than \(\dim V\) or \(\dim W\) then the corresponding summand is zero.

Note that we never used that \(V,W\) were finite dimensional, so the statement is valid for any complex vector spaces \(V,W\).

30 Fundamental and minuscule weights↩︎

30.1 Minuscule weights↩︎

Let \(\mathfrak{g}\) be a simple complex Lie algebra. Minuscule weights for \(\mathfrak{g}\) are highest weights for which irreducible representations are especially simple.

Equivalently, \(|(\omega,\beta)|\le 1\) for any coroot \(\beta\).

Obviously, \(\omega=0\) is minuscule, but there may exist other minuscule weights. For example, for \(\mathfrak{g}=\mathfrak{sl}_n\), all fundamental weights are minuscule, since \((\omega_i, \boldsymbol{e}_j-\boldsymbol{e}_k)=0\) if \(j,k\le i\) or \(j,k>i\) and \((\omega_i, \boldsymbol{e}_j-\boldsymbol{e}_k)=1\) if \(j\le i<k\).

It is easy to see that any minuscule weight \(\omega\ne 0\) is fundamental. Indeed, we can have \((\omega,\alpha_i^\vee)=1\) only for one \(i\), and for all other simple coroots this inner product must be zero. Otherwise we will have \((\omega,\theta^\vee)\ge 2\), where \(\theta^\vee\) is the maximal coroot (the maximal root of the dual root system \(R^\vee\)).¹⁴

On the other hand, not all fundamental weights are minuscule. In fact, we will see that the simple Lie algebras of types \(G_2\), \(F_4\) and \(E_8\) do not have any nonzero minuscule weights. To formulate a criterion for a fundamental weight to be minuscule, recall that \(\theta^\vee=\sum_i k_i\alpha_i^\vee\), where \(k_i=(\omega_i,\theta^\vee)\) are strictly positive integers.

Why are minuscule weights interesting? It is because of the following result.

This immediately implies

30.2 Tensor product with a minuscule representation↩︎

Here in (ii) we sum over all ways to delete a removable box from the Young diagram of \(\mu\), i.e., such that the remaining collection of boxes is still a Young diagram.

Let \(\lambda^\dagger\) be the conjugate partition to \(\lambda\), which consists of the boxes \((j,i)\) where \((i,j)\in \lambda\). In other words, the Young diagram of \(\lambda^\dagger\) is obtained by transposing the Young diagram of \(\lambda\). For example, \((3,3,2,1)^\dagger=(4,3,2)\).

30.3 Fundamental weights of classical Lie algebras↩︎

Let us now determine the fundamental weights of classical Lie algebras of types \(B_n,C_n,D_n\).

Type \(C_n\). Then \(\mathfrak{g}=\mathfrak{sp}_{2n}\). The positive roots are \(\boldsymbol{e}_i\pm \boldsymbol{e}_j,2\boldsymbol{e}_i\), the simple roots \(\alpha_1=\boldsymbol{e}_1-\boldsymbol{e}_2,...,\alpha_n=2\boldsymbol{e}_n\), so \(\alpha_i^\vee=\alpha_i\) for \(i\ne n\) and \(\alpha_n^\vee=\boldsymbol{e}_n\). So \(\omega_i=(1,...,1,0,...,0)\) (\(i\) ones) for \(1\le i\le n\).

Type \(B_n\). Then \(\mathfrak{g}=\mathfrak{so}_{2n+1}\), so we have the same story as for \(C_n\) except \(\alpha_n=\boldsymbol{e}_n\) and \(\alpha_n^\vee=2\boldsymbol{e}_n\), so we have the same \(\omega_i\) for \(i<n\) but \(\omega_n=(\frac{1}{2},...,\frac{1}{2})\).

Type \(D_n\). Then \(\mathfrak{g}=\mathfrak{so}_{2n}\), so the positive roots are \(\boldsymbol{e}_i\pm \boldsymbol{e}_j\), the simple roots \(\alpha_1=\boldsymbol{e}_1-\boldsymbol{e}_2,...,\alpha_{n-2}=\boldsymbol{e}_{n-2}-\boldsymbol{e}_{n-1}\), \(\alpha_{n-1}=\boldsymbol{e}_{n-1}-\boldsymbol{e}_n\), \(\alpha_n=\boldsymbol{e}_{n-1}+\boldsymbol{e}_n\). So \(\omega_i=(1,...,1,0,...,0)\) (\(i\) ones) for \(i=1,...,n-2\), but \(\omega_{n-1}=(\frac{1}{2},...,\frac{1}{2},-\frac{1}{2})\), \(\omega_n=(\frac{1}{2},...,\frac{1}{2},\frac{1}{2})\).

30.4 Minuscule weights outside type \(A\)↩︎

Proposition 194 immediately tells us how many minuscule weights we have. For type \(A\) we saw that all fundamental weights are minuscule. For \(G_2,F_4,E_8\), \(\det A=1\), so the only minuscule weight is \(0\). For type \(B_n\) we have \(\det A=2\), so we should have one nonzero minuscule weight, and this is the weight \((\frac{1}{2},...,\frac{1}{2})\). The corresponding representation has weights \((\pm\frac{1}{2},...,\pm\frac{1}{2})\), so it has dimension \(2^n\). It is called the spin representation, denoted \(S\).

For \(C_n\) we also have \(\det A=2\), so we again have a unique nonzero minuscule weight. Namely, it is the weight \((1,0,...,0)\) (so the minuscule representation is the tautological representation of \(\mathfrak{sp}_{2n}\), of dimension \(2n\)). For \(D_n\) we have \(\det A=4\), so we have three nontrivial minuscule representations, with highest weights \(\omega_1,\omega_{n-1},\omega_n\), of dimensions \(2n,2^{n-1},2^{n-1}\). The first one is the tautological representation and the remaining two are the spin representations \(S_-,S_+\), whose weights are \((\pm\frac{1}{2},...,\pm\frac{1}{2})\) with odd, respectively even number of minuses.

For \(E_6\) there are two nontrivial minuscule representations \(V,V^*\) of dimension \(27\). For \(E_7\) there is just one of dimension \(56\). These dimensions are computed easily by counting elements in the corresponding Weyl group orbits.

31 Fundamental representations of classical Lie algebras↩︎

31.1 Type \(C_n\)↩︎

Since the fundamental weights for \(\mathfrak{g}=\mathfrak{sp}_{2n}\) are \(\omega_i=(1,...,1,0,..,0)\) (\(i\) ones), same as for \(\mathfrak{gl}_n\), one may think that the fundamental representations are also “the same", i.e. \(\wedge^i V\), where \(V\) is the \(2n\)-dimensional vector representation. Indeed, a Cartan subalgebra in \(\mathfrak{g}\) is the space of matrices \({\rm diag}(a_1,...,a_n,-a_1,...,-a_n)\), so \(L_{\omega_1}=V\), with highest weight vector \(e_1\). However, the representation \(\wedge^2V\) is not irreducible, even though it has the correct highest weight \(\omega_2\). Indeed, we have \(\wedge^2V=\wedge^2_0V\oplus \Bbb C\), where \(\Bbb C\) is the trivial representation spanned by the inverse \(B^{-1}=\sum_i e_{i+n}\wedge e_i\) of the invariant nondegenerate skew-symmetric form \(B=\sum_i e_i^*\wedge e_{i+n}^*\in \wedge^2V^*\) preserved by \(\mathfrak{g}\), and \(\wedge^2_0V\) is the orthogonal complement of \(B\).

It turns out that \(\wedge^2_0V\) is irreducible. (You can show it directly or using the Weyl dimension formula). Thus we have \(L_{\omega_2}=\wedge^2_0V\) (if \(n\ge 2\)).

So what happens for \(L_{\omega_j}\) with any \(j\ge 2\)? To determine this, note that we have a homomorphism of representations \(\iota_B: \wedge^{i} V\to \wedge^{i-2}V\), which is just the contraction with \(B\) (we agree that \(\wedge^jV=0\) for \(j<0\)). So we may consider the subrepresentation \(\wedge^i_0V={\rm Ker}(\iota_B|_{\wedge^iV})\subset \wedge^iV\).

31.2 Type \(B_n\)↩︎

We have \(\mathfrak{g}=\mathfrak{so}_{2n+1}\), preserving the quadratic form \(Q=\sum_{i=1}^n x_ix_{i+n}+x_{2n+1}^2\). A Cartan subalgebra consists of matrices \({\rm diag}(a_1,...,a_n,-a_1,...,-a_n,0)\). So the representations \(\wedge^iV\), \(1\le i\le n\), where \(V\) is the \(2n+1\)-dimensional vector representation, have highest weight \((1,...,1,0,...0)\) (\(i\) ones), which is \(\omega_i\) if \(i\le n-1\).

Thus for \(1\le i\le n-1\) we have \(\wedge^i V=L_{\omega_i}\). On the other hand, the representation \(\wedge^nV\), even though irreducible, is not fundamental. Indeed, its highest weight is \((1,...,1)=2\omega_n\), as \(\omega_n=(\frac{1}{2},...,\frac{1}{2})\). In fact, we see that the representation \(L_{\omega_n}\) does not occur in \(V^{\otimes N}\) for any \(N\), since coordinates of its highest weight are not integer. As mentioned above, this representation is called the spin representation \(S\). Vectors in \(S\) are called spinors. The weights of \(S\) are Weyl group translates of \(\omega_n\), so they are \((\pm\frac{1}{2},...,\pm\frac{1}{2})\) for any choices of signs, so \(\dim S=2^n\), and the character of \(S\) is given by the formula \[\chi_S(x_1,...,x_n)=(x_1^{\frac{1}{2}}+x_1^{-\frac{1}{2}})...(x_n^{\frac{1}{2}}+x_n^{-\frac{1}{2}}).\] This is supposed to be the trace of \(\text{diag}(x_1,...,x_n,x_1^{-1},...,x_n^{-1},1)\in SO_{2n+1}(\Bbb C)\), which does not make sense since the square roots on the right hand side are defined only up to sign. This shows that the spin representation \(S\) does not lift to the group \(SO_{2n+1}(\Bbb C)\). Namely, the group \(SO_{2n+1}(\Bbb C)\) is not simply connected, and the representation \(S\) only lifts to the universal covering group \(\widetilde{SO}_{2n+1}(\Bbb C)\), which is called the spin group, and is denoted \({\rm Spin}_{2n+1}(\Bbb C)\).

31.3 Type \(D_n\)↩︎

We have \(\mathfrak{g}=\mathfrak{so}_{2n}\), preserving the quadratic form \[Q=\sum_{i=1}^n x_ix_{i+n}.\] A Cartan subalgebra consists of matrices \({\rm diag}(a_1,...,a_n,-a_1,...,-a_n)\). So the representations \(\wedge^iV\), \(1\le i\le n\), where \(V\) is the \(2n\)-dimensional vector representation, have highest weight \((1,...,1,0,...0)\) (\(i\) ones), which is \(\omega_i\) if \(i\le n-2\).

Thus \(L_{\omega_i}=\wedge^i V\) for \(i\le n-2\). On the other hand, while the representation \(L_{(1,...,1,0)}\) is irreducible, it is not fundamental, as \((1,...,1,0)=\omega_{n-1}+\omega_n\), where \(\omega_{n-1}=(\frac{1}{2},...,\frac{1}{2}, -\frac{1}{2})\) and \(\omega_n=(\frac{1}{2},...,\frac{1}{2}, \frac{1}{2})\). The fundamental representations \(L_{\omega_{n-1}},L_{\omega_n}\) are called the spin representations and denoted \(S_-,S_+\); their elements are called spinors. Similarly to the odd dimensional case, they have dimensions \(2^{n-1}\) and characters \[\chi_{S_\pm}=\left((x_1^{\frac{1}{2}}+x_1^{-\frac{1}{2}})...(x_n^{\frac{1}{2}}+x_n^{-\frac{1}{2}})\right)_\pm\] where the subscript \(\pm\) means that we take the monomials with odd (for –), respectively even (for +) number of minuses. This shows that, similarly to the odd dimensional case, \(S_+,S_-\) don’t occur in \(V^{\otimes N}\) and don’t lift to \(SO_{2n}(\Bbb C)\) but require the universal covering \({\rm Spin}_{2n}(\Bbb C)=\widetilde{SO}_{2n}(\Bbb C)\), called the spin group. Proposition 197 implies

31.4 The Clifford algebra↩︎

It is important to be able to realize the spin representations explicitly. The reason it is somewhat tricky is that these representations don’t occur in tensor powers of \(V\) (as they have half-integer weights). However, the tensor product of a spin representation with its dual, \(S\otimes S^*\), has integer weights and does express in terms of \(V\). So we need to extract "the square root" from this representation, in the sense that “the space of vectors of size \(n\) is the square root of the space of square matrices of size \(n\)". This is the idea behind the Clifford algebra construction.

Thus for \(a,b\in V\) we have \[ab+ba=(a+b)^2-a^2-b^2=\tfrac{1}{2}((a+b,a+b)-(a,a)-(b,b))=(a,b).\] This is a deformation of the exterior algebra \(\wedge V\) which is defined in the same way but \(v^2=0\). More precisely, \({\rm Cl}(V)\) has a filtration (defined by setting \(\deg(v)=1\), \(v\in V\)) such that the associated graded algebra receives a surjective map \(\phi: \wedge V\to {\rm gr}{\rm Cl}(V)\). We will show that this is a nice (“flat") deformation, in the sense that \(\dim {\rm Cl}(V)=\dim \wedge V=2^{\dim V}\), so that \(\phi\) is an isomorphism. This is a kind of Poincaré-Birkhoff-Witt theorem (namely, it is similar to the PBW theorem for Lie algebras, and in fact a special case of one if you pass from Lie algebras to more general Lie superalgebras). Namely, we have the following theorem.

We will now construct an inclusion of the Lie algebra \(\mathfrak{so}(V)\) into the Clifford algebra. This will allow us to regard representations of the Clifford algebra as representations of \(\mathfrak{so}(V)\), which will give us a construction of the spin representations.

Consider the linear map \(\xi: \wedge^2V=\mathfrak{so}(V)\to {\rm Cl}(V)\) given by the formula \[\xi(a\wedge b)=\tfrac{1}{2}(ab-ba)=ab-\tfrac{1}{2}(a,b).\] Then \[[\xi(a\wedge b),\xi(c\wedge d)]=[ab,cd]=abcd-cdab=(b,c)ad-acbd-cdab=\] \[(b,c)ad-(b,d)ac+acdb-cdab=\] \[(b,c)ad-(b,d)ac+(a,c)db-cadb-cdab=\] \[(b,c)ad-(b,d)ac+(a,c)db-(a,d)cb=\] \[(b,c)\xi(a\wedge d)-(b,d)\xi(a\wedge c)+(a,c)\xi(d\wedge b)-(a,d)\xi(c\wedge b)= \xi([a\wedge b,c\wedge d]).\] Thus \(\xi\) is a homomorphism of Lie algebras and we can define the representations \(\xi^*M\) for even \(\dim V\) and \(\xi^*M_{\pm}\) for odd \(\dim V\) by \(\rho_{\xi^*M}(a):=\rho_M(\xi(a))\).

The representation \(\xi^*M\) is reducible, namely \[\xi^*M=(\xi^*M)_0\oplus (\xi^*M)_1,\] where subscripts \(0\) and \(1\) indicate the even and odd degree parts.

32 Maximal root, exponents, Coxeter numbers, dual representations↩︎

32.1 Duals of irreducible representations↩︎

Now let \(\mathfrak{g}\) be any complex semisimple Lie algebra. How to compute the dual of the irreducible representation \(L_\lambda\)? It is clear that the highest weight of \(L_\lambda^*\) equals \(-\mu\), where \(\mu\) is the lowest weight of \(L_\lambda\), so we should compute the latter. For this purpose, recall that the Weyl group \(W\) of \(\mathfrak{g}\) contains a unique element \(w_0\) which maps dominant weights to antidominant weights, i.e., maps positive roots to negative roots. This is the maximal element, which is the unique element whose length is \(|R_+|\). For example, if \(-1\in W\) then clearly \(w_0=-1\). It is easy to see that the lowest weight of \(L_\lambda\) is \(w_0\lambda\).

Thus we get

The map \(-w_0\) permutes fundamental (co)weights and simple (co)roots, so it is induced by an automorphism of the Dynkin diagram of \(\mathfrak{g}\). So if \(\mathfrak{g}\) is simple and its Dynkin diagram has no nontrivial automorphisms, we have \(w_0=-1\), so \(-w_0=1\) and thus \(L_\lambda^*=L_\lambda\) for all \(\lambda\). This happens for \(A_1\), \(B_n\), \(C_n\), \(G_2\), \(F_4\), \(E_7\) and \(E_8\). In general, note that \(s_i\) and hence the whole Weyl group \(W\) acts trivially on \(P/Q\), which implies that \(-w_0\) acts on \(P/Q\) by inversion. Thus we see that for \(A_{n-1}\), \(n\ge 3\), when \(P/Q=\Bbb Z/n\), the map \(-w_0\) is the flip of the chain. Another way to see it is to note that \(L_{\omega_1}^*=V^*=\wedge^{n-1}V=L_{\omega_{n-1}}\) (as \(\dim V=n\)). For \(E_6\), \(P/Q=\Bbb Z/3\), so \(-w_0\) must exchange the two nonzero minuscule weights and thus must also be the flip.

32.2 The maximal root↩︎

Let \(\mathfrak{g}\) be a complex simple Lie algebra and \(\theta\) be the maximal root of \(\mathfrak{g}\), i.e., the highest weight of the adjoint representation. For example, for \(\mathfrak{g}=\mathfrak{sl}_n\) the adjoint representation is generated by the highest weight vector of \(V\otimes V^*\), where \(V=\Bbb C^n\) is the vector representation. Thus we have \[\theta=\omega_1+\omega_{n-1}=(2,1,...,1,0)=(1,0,...,0,-1),\] the sum of the highest weights of \(V\) and \(V^*\) (recall that weights for \(\mathfrak{sl}_n\) are \(n\)-tuples of complex numbers modulo simultaneous translation by the same number). Thus, \(\theta\) is not fundamental. Similarly, for \(\mathfrak{g}=\mathfrak{sp}_{2n}\), we have \(\mathfrak{g}=S^2V\) where \(V\) is the vector representation, so \(\theta=2\omega_1\) is again not fundamental. Nevertheless, we have the following proposition.

32.3 Principal \(\mathfrak{sl}_2\), exponents↩︎

Let \(\mathfrak{g}\) be a simple Lie algebra and let \(e=\sum_i e_i\) and \(h\in \mathfrak{h}\) be such that \(\alpha_i(h)=2\) for all \(i\) (i.e., \(h=2\rho^\vee\)). We have \([h,e]=2e\) and \(h=\sum_i (2\rho^\vee,\omega_i)h_i\). So defining \(f:=\sum_i (2\rho^\vee,\omega_i)f_i\), we have \([h,f]=-2f\), \([e,f]=h\). So \(e,f,h\) span an \(\mathfrak{sl}_2\)-subalgebra of \(\mathfrak{g}\) called the principal \(\mathfrak{sl}_2\)-subalgebra.

Consider now \(\mathfrak{g}\) as a module over its principal \(\mathfrak{sl}_2\)-subalgebra. How does it decompose? To see this, we can look at the weight decomposition of \(\mathfrak{g}\) under \(h\). We have \(\mathfrak{g}=\mathfrak{n}_-\oplus \mathfrak{h}\oplus \mathfrak{n}_+\), and these summands correspond to negative, zero and positive weights, respectively. Moreover, all weights are even, and for \(m>0\), \(\dim \mathfrak{g}[2m]=r_m\) is the number of positive roots of height \(m\), i.e., representable as a sum of \(m\) simple roots, while \(\mathfrak{g}[0]=\mathfrak{h}\) (as \(\rho^\vee\) is a regular coweight), so \(\dim \mathfrak{g}[0]=r\), the rank of \(\mathfrak{g}\).

Since \(r_m\) is zero for large \(m\) while \(r_0=r\), there are \(r\) exponents counting multiplicities. The exponents of \(\mathfrak{g}\) are denoted \(m_i\) and are arranged in non-decreasing order: \(m_1\le m_2\le...\le m_r\) (including multiplicities). Note that roots of height \(2\) are \(\alpha_i+\alpha_j\) where \(i,j\) are connected by an edge. Thus we have \(r_0=r_1=r\), \(r_2=r-1\) (as the Dynkin diagram of \(\mathfrak{g}\) is a tree), so \(m_1=1\) and \(m_2>1\). We also have \(m_r=(\rho^\vee,\theta):={\rm h}_\mathfrak{g}-1\), where \(\theta\) is the maximal root. The number \({\rm h}_\mathfrak{g}\) is called the Coxeter number of \(\mathfrak{g}\). Finally, we have \(\sum_{i=1}^r m_i=|R_+|\).

32.4 The Coxeter number and the dual Coxeter number↩︎

We have defined the Coxeter number of a simple complex Lie algebra \(\mathfrak{g}\) (or a reduced irreducible root system \(R\)) to be \({\rm h}_R={\rm h}_\mathfrak{g}:=(\theta,\rho^\vee)+1=m_r+1\), where \(m_r\) is the largest exponent of \(\mathfrak{g}\). One can also define the dual Coxeter number of \(\mathfrak{g}\) (or \(R\)) as \({\rm h}_R^\vee={\rm h}_\mathfrak{g}^\vee:=(\widetilde{\theta}^\vee,\rho)+1\), cf. footnote 14 (clearly, \({\rm h}_R^\vee={\rm h}_R\) if \(R\) is simply laced). So the dual Coxeter number is the eigenvalue \(\frac{1}{2}(\theta,\theta+2\rho)\) of \(\frac{1}{2}C\) on the adjoint representation \(\mathfrak{g}\), where \(C\in U(\mathfrak{g})\) is the quadratic Casimir element defined using the inner product in which \((\theta,\theta)=2\) (or, equivalently, long roots have squared length \(2\)). Indeed, if we identify \(\mathfrak{h}\) and \(\mathfrak{h}^*\) using this inner product then \(\theta\) gets identified with \(\widetilde{\theta}^\vee\).

Using the formulas from Subsections 23.7 and 32.2, we get \[{\rm h}_{A_{n-1}}=n,\] \[{\rm h}_{B_n}=2n,\;{\rm h}_{B_n}^\vee=2n-1,\] \[{\rm h}_{C_n}=2n,\;{\rm h}_{C_n}^\vee=n+1,\] \[{\rm h}_{D_n}=2n-2,\] \[{\rm h}_{G_2}=(2\alpha+3\beta,5\alpha^\vee+3\beta^\vee)+1=6,\; h_{G_2}^\vee=\frac{1}{3}(2\alpha+3\beta,3\alpha+5\beta)+1=4,\] \[{\rm h}_{F_4}=(8,3,2,1)\cdot (1,1,0,0)+1=12,\; {\rm h}_{F_4}^\vee=(\tfrac{11}{2},\tfrac{5}{2},\tfrac{3}{2},\tfrac{1}{2})\cdot (1,1,0,0)+1=9,\] \[{\rm h}_{E_8}=(23,6,5,4,3,2,1,0)\cdot (1,1,0,0,0,0,0,0)+1=30,\] \[{\rm h}_{E_7}=(\tfrac{17}{2},-\tfrac{17}{2},5,4,3,2,1,0)\cdot (1,-1,0,0,0,0,0,0)+1=18,\] \[{\rm h}_{E_6}=(4,-4,-4,4,3,2,1,0)\cdot \tfrac{1}{2}(1,-1,-1,1,1,1,1,1)+1=12.\] Note that we always have \({\rm h}_R={\rm h}_{R^{\vee}}\), but if \(R\) is not simply laced then, as we see, the numbers \({\rm h}_R\), \({\rm h}_{R^\vee}^\vee\), \({\rm h}_R^\vee\) are different, in general.

32.5 Representations of complex, real and quaternionic type↩︎

It is easy to see that any irreducible finite dimensional representation is of exactly one of these three types (check it!).

Now let \(\mathfrak{g}\) be any simple Lie algebra and \(\lambda\in P_+\) be such that \(\lambda=-w_0\lambda\), so that \(L_\lambda\) is selfdual. How to tell if it is of real or quaternionic type?

We obtain the following result.

33 Differential forms, partitions of unity↩︎

Now we want to develop an integration theory on Lie groups. First we need to recall the basics about integration on manifolds.

33.1 Locally compact spaces↩︎

A Hausdorff topological space \(X\) is called locally compact if every point has a neighborhood whose closure is compact. For example, \(\Bbb R^n\) and thus every manifold is locally compact.

An open cover of a topological space \(X\) is said to be locally finite if every point of \(X\) has a neighborhood intersecting only finitely many members of this cover.

33.2 Reminder on differential forms↩︎

Let \(M\) be a real smooth \(n\)-dimensional manifold. Recall that a differential \(k\)-form on \(M\) is a smooth section of the vector bundle \(\wedge^kT^*M\), i.e., a skew-symmetric \((0,k)\)-tensor field (see Subsection 5.3). Thus, for example, a 1-form is a section of \(T^*M\). If \(x_1,...,x_n\) are local coordinates on \(M\) near some point \(p\in M\) then the differentials \(dx_1,...,dx_n\) form a basis in fibers of \(T^*M\) near this point, so a general \(1\)-form in these coordinates has the form \[\omega=\sum_{i=1}^nf_i(x_1,...,x_n)dx_i.\] If we change the coordinates \(x_1,...,x_n\) to \(y_1,...,y_n\) then \(x_i\) are smooth functions of \(y_1,...,y_n\) and in the new coordinates \(\omega\) looks like \[\omega=\sum_{i,j=1}^n f_i(x_1,...,x_n)\frac{\partial x_i}{\partial y_j}dy_j.\] Similarly, a differential \(k\)-form in the coordinates \(x_i\) looks like \[\omega=\sum_{1\le i_1<...< i_k\le n}f_{i_1,...,i_k}(x_1,...,x_n)dx_{i_1}\wedge...\wedge dx_{i_k}\] where \(f_{i_1,...,i_k}\) are smooth functions, and in the coordinates \(y_j\) it looks like \[\omega=\sum_{1\le i_1<...<i_k\le n}\sum_{1\le j_1<...<j_k\le n}f_{i_1,...,i_k}(x_1,...,x_n)\det\left(\frac{\partial x_{i_r}}{\partial y_{j_s}}\right)dy_{j_1}\wedge...\wedge dy_{j_k}.\] The space of differential \(k\)-forms on \(M\) is denoted \(\Omega^k(M)\). For instance, \(\Omega^0(M)=C^\infty(M)\) and \(\Omega^k(M)=0\) for \(k>n\). Consider now the extremal case \(k=n\). The bundle \(\wedge^n T^*M\) is a line bundle (a vector bundle of rank \(1\)), so locally any differential \(n\)-form in coordinates \(x_i\) has the form \[\omega=f(x_1,...,x_n)dx_1\wedge...\wedge dx_n,\] which in coordinates \(y_j\) takes the form \[\omega=f(x_1,...,x_n)\det\left(\frac{\partial x_i}{\partial y_j}\right)dy_1\wedge...\wedge dy_n.\]

We have a canonical differentiation operator \(d: \Omega^0(M)\to \Omega^1(M)\) given in local coordinates by \[df=\sum_{i=1}^n \frac{\partial f}{\partial x_i}dx_i.\] It is easy to check that this operator does not depend on the choice of coordinates (this becomes obvious if you define it without coordinates, \(df(v)=\partial_v f\) for \(v\in T_pM\)). Also \(\Omega^\bullet(M):=\oplus_{k=0}^n \Omega^k(M)\) is a graded algebra under wedge product, and \(d\) naturally extends to a degree \(1\) derivation \(d: \Omega^\bullet(M)\to \Omega^\bullet (M)\) defined in coordinates by \[d(fdx_{i_1}\wedge...\wedge dx_{i_k})=df\wedge dx_{i_1}\wedge ...\wedge dx_{i_k}.\] Namely, this is independent of choices and gives rise to a derivation in the “graded" sense: \[d(a\wedge b)=da\wedge b+(-1)^{\deg a}a\wedge db.\] A form \(\omega\) is closed if \(d\omega=0\) and exact if \(\omega=d\eta\) for some \(\eta\). It is easy to check that \(d^2=0\), so any exact form is closed. However, not every closed form is exact: on the circle \(S^1=\Bbb R/\Bbb Z\) the form \(dx\) is closed but the function \(x\) is defined only up to adding integers, so \(dx\) is not exact. The space \(\Omega^k_{\rm closed}(M)/\Omega^k_{\rm exact}(M)\) is called the \(k\)-th de Rham cohomology of \(M\), denoted \(H^k(M)\).

If \(f: M\to N\) is a differentiable mapping then for a differential form \(\omega\in \Omega^k(N)\) we can define the pullback \(f^*\omega\in \Omega^k(M)\), given by \((f^*\omega)(v_1,...,v_k)=\omega(f_*v_1,...,f_*v_k)\) for \(v_1,...,v_k\in T_pM\). This operation commutes with wedge product and the differential, and \((f\circ g)^*=g^*\circ f^*\).

33.3 Partitions of unity↩︎

Let \(M\) be a manifold and \(\lbrace{U_i, i\in I\rbrace}\) be an open cover of \(M\).

Note that the sum in (iii) makes sense because of condition (ii).

Note also that given any partition of unity \(\lbrace f_s\rbrace\) subordinate to \(\lbrace U_i\rbrace\), we can define \[F_i:=\sum_{s: i(s)=i}f_s,\] and this is a new partition of unity subordinate to the same cover now labeled by the set \(I\), with the support of \(F_i\) contained in \(U_i\).

Finally, note that in every partition of unity on \(M\), the set of \(s\) such that \(f_s\) is not identically zero is countable, and moreover finite if \(M\) is compact. This follows from the fact that by Lemma 1, any open cover of a manifold \(M\) has a countable subcover, and moreover a finite one if \(M\) is compact (applied to the neighborhoods from condition (ii)).

34 Integration on manifolds↩︎

34.1 Integration of top differential forms on oriented manifolds↩︎

An important operation with top degree differential forms is integration. Namely, if \(\omega\) is a differential \(n\)-form on an open set \(U\subset \Bbb R^n\) (with the usual orientation), \(\omega=f(x_1,...,x_n)dx_1\wedge...\wedge dx_n\), then we can set \[\int_{U}\omega:=\int_{U}f(x_1,...,x_n)dx_1...dx_n.\] (provided this integral is absolutely convergent). This, however, is not completely canonical: if we change coordinates (so that \(U\) maps diffeomorphically to \(U'\)), the change of variable formula in a multiple integral tells us that \[\int_{U}f(x_1,...,x_n)dx_1\wedge...\wedge dx_n=\int_{U'} f(x_1(\boldsymbol{y}),...,x_n(\boldsymbol{y}))\bigg|\det\left(\tfrac{\partial x_i}{\partial y_j}\right)\bigg| dy_1\wedge...\wedge dy_n,\] while the transformation law for \(\omega\) is the same but without the absolute value. This shows that our definition is invariant only under orientation preserving transformations of coordinates, i.e., ones whose Jacobian \(\det\left(\frac{\partial x_i}{\partial y_j}\right)\) is positive. Consequently, we will only be able to define integration of top differential forms on oriented manifolds, i.e., ones equipped with an atlas of charts in which transition maps have a positive Jacobian; such an atlas defines an orientation on \(M\). To fix an orientation, we just need to say which local coordinate systems (or bases of tangent spaces) are right-handed, and do so in a consistent way. But this cannot always be done globally (the classic counterexamples are Möbius strip and Klein bottle).

Now let us proceed to define integration of a continuous top form \(\omega\) over an oriented manifold \(M\). For this pick an atlas of local charts \(\lbrace U_i,i\in I\rbrace\) on \(M\) and pick a partition of unity \(\lbrace f_s\rbrace\) subordinate to this cover, which is possible by Proposition 215. First assume that \(\omega\) is nonnegative, i.e., \(\omega(v_1,...,v_n)\ge 0\) for a right-handed basis \(v_i\) of any tangent space of \(M\). Then define \[\label{defint} \int_M \omega:=\sum_{s} \int_{U_{i(s)}}f_s\omega\tag{11}\] where in each \(U_i\) we use a right-handed coordinate system to compute the corresponding integral. This makes sense (as a nonnegative real number or \(+\infty\)), and is also independent of the choice of a partition of unity. Indeed, it is easy to see that for two atlases \(\lbrace U_i\rbrace\), \(\lbrace V_j\rbrace\) and two partitions of unity \(\lbrace f_s\rbrace ,\lbrace g_t\rbrace\) the answer is the same, by comparing both to the answer for the atlas \(\lbrace U_i\cap V_j\rbrace\) and partition of unity \(\lbrace f_sg_t\rbrace\). In fact, this makes sense for any measurable \(\omega\) (i.e., given by a measurable function in every local chart) if we use Lebesgue integration.

Now, if \(\omega\) is not necessarily nonnegative, we may define the nonnegative form \(|\omega|\) which is \(\omega\) at points where \(\omega\) is nonnegative and \(-\omega\) otherwise. Then, if \[\int_M|\omega|<\infty,\] we can define \(\int_M\omega\) by the same formula 11 which will now be a not necessarily positive but absolutely convergent series (a finite sum in the compact case).

Importantly, the same definition works for manifolds \(M\) with boundary \(\partial M\) (an \(n-1\)-manifold); the only difference is that at boundary points the manifold locally looks like \(\Bbb R^n_+\) (the space of vectors with nonnegative last coordinate) rather than \(\Bbb R^n\). Note that the boundary of an oriented manifold carries a canonical orientation as well (a basis of \(T_p\partial M\) is right-handed if adding at the beginning a tangent vector directed outside \(M\) produces a right-handed basis of \(T_pM\)).

34.2 Nonvanishing forms↩︎

Let us say that a top degree continuous differential form \(\omega\) on \(M\) is non-vanishing if for any \(x\in M\), \(\omega_x\in \wedge^nT_x^*M\) is nonzero. In this case, \(\omega\) defines an orientation on \(M\) by declaring a basis \(v_1,...,v_n\) of \(T_xM\) right-handed if \(\omega(v_1,...,v_n)>0\) (in particular, there are no non-vanishing top forms on non-orientable manifolds). Thus we can integrate top differential forms on \(M\), and in particular \(\omega\) defines a positive measure \(\mu=\mu_\omega\) on \(M\), namely \[\mu(U)=\int_U\omega\] for an open set \(U\subset M\) (this integral may be \(+\infty\), but is finite if \(U\) is a small enough neighborhood of any point \(x\in M\)). Thus we can integrate functions on \(M\) with respect to this measure: \[\int_M fd\mu=\int_M f\omega.\] This, of course, only makes sense if \(f\) is measurable and \(\int_M |f|d\mu<\infty\), i.e., if \(f\in L^1(M,\mu)\). Note also that if \(\lambda\in \Bbb R^\times\) then \(\mu_{\lambda\omega}=|\lambda|\mu_\omega\).

34.3 Stokes formula↩︎

A central result about integration of differential forms is

In particular, if \(M\) is closed (has no boundary) then \(\int_M d\omega=0\), and if \(\omega\) is closed (\(d\omega=0\)) then \(\int_{\partial M}\omega=0\).

When \(M\) is an interval in \(\Bbb R\), this reduces to the fundamental theorem of calculus. If \(M\) is a region in \(\Bbb R^2\), this reduces to Green’s formula. If \(M\) is a surface in \(\Bbb R^3\), this reduces to the classical Stokes formula from vector calculus. Finally, if \(M\) is a region in \(\Bbb R^3\) then this reduces to the Gauss formula (Divergence theorem).

The proof of the Stokes formula is not difficult. Namely, by writing \(\omega\) as \(\sum_s f_s\omega\) for some partition of unity, it suffices to prove the formula for \(M\) being a box in \(\Bbb R^n\), which easily follows from the fundamental theorem of calculus.

34.4 Integration on Lie groups↩︎

Now let \(G\) be a real Lie group of dimension \(n\). In this case given any \(\xi\in \wedge^n\mathfrak{g}^*\), we can extend it to a left-invariant skew-symmetric tensor field (i.e., top differential form) \(\omega_\xi\) on \(G\). Also, if \(\xi\ne 0\) then \(\omega=\omega_\xi\) is non-vanishing and thus defines an orientation and a left-invariant positive measure \(\mu_\omega\) on \(G\). Note that \(\xi\) is unique up to scaling by a real number \(\lambda\in \Bbb R^\times\). So, since \(\mu_{\lambda \omega}=|\lambda|\mu_\omega\), we see that \(\mu_\omega\) is defined uniquely up to scaling by positive numbers. This measure is called the left-invariant Haar measure and we’ll denote it just by \(\mu_L\) (assuming that the normalization has been chosen somehow).

In a similar way we can define the right invariant Haar measure \(\mu_R\) on \(G\). One may ask if these measures coincide (or, rather, are proportional, since they are defined only up to normalization). This question is answered by the following proposition.

Given a 1-dimensional real representation \(V\) of a group \(G\), let \(|V|\) be the representation of \(G\) on the same space with \(\rho_{|V|}(g)=|\rho_V(g)|\), where \(\rho: G\to {\rm Aut}(V)=\Bbb R^\times\).

If \(\mu_L=\mu_R\) then \(G\) is called unimodular. In this case we have a bi-invariant Haar measure \(\mu=\mu_L=\mu_R\) on \(G\) (under some normalization).

In particular, we see that if \(G\) has no nontrivial continuous characters \(G\to \Bbb R^+\) then it is unimodular.

For a unimodular Lie group \(G\), we will sometimes denote the integral of a function \(f\) with respect to the Haar measure by \[\int_G f(g)dg.\]

Thus, on a compact Lie group we have a (bi-invariant) Haar measure \(\mu\). Moreover, in this case \(\int_Gd\mu={\rm Volume}(G)<\infty\), so we have a canonical normalization of \(\mu\) by the condition that it is a probability measure: \[\int_G d\mu=1.\] E.g., for finite groups this normalization is the averaging measure, which is \(|G|^{-1}\) times the counting measure. This is the normalization we will use if \(G\) is compact.

35 Representations of compact Lie groups↩︎

35.1 Unitary representations↩︎

Now we can extend to compact groups the result that representations of finite groups are unitary. Namely, let \(V\) be a finite dimensional (continuous) complex representation of a compact Lie group \(G\).

In particular, this applies to the special unitary group \(SU(n)\). Recall that \(SU(n)/SU(n-1)=S^{2n-1}\), which implies that \(SU(n)\) is simply connected. Thus (smooth) representations of \(SU(n)\) are the same thing as representations of the Lie algebra \(\mathfrak{su}(n)\) or its complexification \(\mathfrak{sl}_n\). Thus we get a new, analytic proof that finite dimensional representations of \(\mathfrak{sl}_n\) are completely reducible (this is called Weyl’s unitary trick). In fact, we will see that complete reducibility of finite dimensional representations of all semisimple Lie algebras can be proved in this way.

35.2 Matrix coefficients↩︎

Let \(V\) be a finite dimensional continuous complex representation of a Lie group \(G\). A matrix coefficient of \(V\) is a function \(G\to \Bbb C\) of the form \((f,\rho_V(g)v)\) for some \(v\in V\) and \(f\in V^*\). Obviously, such a function is continuous.

Now let \(V\) be an irreducible representation of a compact Lie group \(G\). As shown above, it has an invariant positive Hermitian inner product, which we’ll denote by \((,)\). Moreover, this product is unique up to scaling. Pick an orthonormal basis \(v_1,...,v_n\) of \(V\) under this inner product, and let \(v_1^*,...,v_n^*\) be the dual basis of \(V^*\). Now consider the matrix coefficients of \(V\) in this basis: \[\psi_{V,ij}(g):=v_j^*(\rho_V(g)v_i)=(\rho_V(g)v_i,v_j).\] Note that these functions are independent of the normalization of \((,)\).

Suppose now that we also have another such representation \(W\) with orthonormal basis \(w_i\).

35.3 The Peter-Weyl theorem↩︎

Thus we see that the functions \(\psi_{V,ij}\) for various \(V,i,j\) form an orthogonal system in the Hilbert space \(L^2(G)=L^2(G,dg)\) of measurable functions \(f: G\to \Bbb C\) such that \[||f||^2=\int_G |f(g)|^2dg<\infty.\] A fundamental result about compact Lie groups is that this system is, in fact, complete:

Theorem 225 will be proved in Section 36.

35.4 An alternative formulation of the Peter-Weyl theorem↩︎

Given a finite dimensional irreducible representation \(V\) of \(G\), consider the space \(\mathrm{Hom}_G(V,L^2(G))\) of \(G\)-homomorphisms for the action of \(G\) on \(L^2(G)\) by left translations. We have an obvious inclusion \[\iota_V: V^*\hookrightarrow \mathrm{Hom}_G(V,L^2(G))\] via the matrix coefficient map \(f\mapsto [v\mapsto (\rho_{V^*}(-)f)(v)]\). Clearly, this is a map of \(G\)-modules, where now \(G\) acts on \(L^2(G)\) by right translations. We claim that \(\iota_V\) is surjective, i.e., an isomorphism. For this, note that an element \(\phi\in \mathrm{Hom}_G(V,L^2(G))\) can be viewed a left \(G\)-equivariant \(L^2\)-function \(\phi: G\to V^*\), i.e. such that for almost all \(g\in G\) (with respect to the Haar measure) we have \[\label{invaa} \phi(x)=\rho_{V^*}(xg^{-1})\phi(g)\tag{12}\] for almost all \(x\in G\). But then by changing \(\phi\) on a set of measure zero if needed, we may replace it by a continuous function (the right hand side of 12 ). Then, setting \(g=1\), we have \(\phi(x)=\rho_{V^*}(x)\phi(1)\), as claimed.

Let \[\iota=\oplus_{V\in {\rm Irrep}(G)}\sqrt{\dim V}{\rm Id}_V\otimes \iota_V: \oplus_{V\in {\rm Irrep}(G)}V\otimes V^*\to \oplus_{V\in {\rm Irrep}(G)}V\otimes \mathrm{Hom}_G(V,L^2(G)).\] Then \(\iota\) defines an isometric embedding of \(G\times G\)-modules \[\xi: \oplus_{V\in {\rm Irrep}(G)}V\otimes V^*\hookrightarrow L^2(G).\] We will denote the image of \(\xi\) by \(L^2_{\rm alg}(G)\) (the “algebraic part" of \(L^2(G)\)). Note that if \(\psi\in L^2(G)\) generates a finite dimensional representation \(V\) under the action of \(G\) by left translations then \(\psi\) belongs to the image of a homomorphism \(V\to L^2(G)\), hence to \(L^2_{\rm alg}(G)\). Thus \(L^2_{\rm alg}(G)\) is just the subspace of \(\psi\in L^2(G)\) which generate a finite dimensional representation under left translations by \(G\). We also see that it may be equivalently characterized as the subspace of \(\psi\in L^2(G)\) which generate a finite dimensional representation under right translations by \(G\).

Note that this is again an instance of the double centralizer property! Namely, it expresses representation-theoretically the fact that the centralizer of the group of left translations on \(G\) is the group of right translations on \(G\), and vice versa.

For example, let \(G=S^1\). Then the irreducible representations of \(G\) are the characters \(\psi_n(\theta)=e^{in\theta}\). So the Peter-Weyl theorem in this case says that \(\lbrace e^{in\theta}\rbrace\) is an orthonormal basis of \(L^2(S^1)\) with norm \[||f||^2:=\frac{1}{2\pi}\int_0^{2\pi}|f(\theta)|^2d\theta,\] which is the starting point for Fourier analysis. So the Peter-Weyl theorem is similarly a starting point for nonabelian Fourier (or harmonic) analysis.

35.5 Orthogonality and completeness of characters↩︎

36 Proof of the Peter-Weyl theorem↩︎

36.1 Compact operators and the Hilbert-Schmidt theorem↩︎

To prove the Peter-Weyl theorem, we will use the Hilbert-Schmidt theorem – the spectral theorem for compact self-adjoint operators in a Hilbert space.

Recall that a bounded operator \(A: H\to H\) on a Hilbert space \(H\) is a linear operator such that for some \(C\ge 0\) we have \(||A\boldsymbol{v}||\le C||\boldsymbol{v}||\), \(\boldsymbol{v}\in H\). The smallest constant \(C\) with this property is called the norm of \(A\) and denoted \(||A||\). Recall also that \(A\) is compact if there is a sequence of finite rank operators \(A_n: H\to H\) such that \(||A_n-A||\to 0\) as \(n\to \infty\). In other words, the space \(K(H)\) of compact operators on \(H\) is the closure of the space \(K_f(H)\) of finite rank operators under the norm \(A\mapsto ||A||\) on the space of bounded operators \(B(H)\).

Recall that a bounded operator \(A\) is self-adjoint if \((A\boldsymbol{v},\boldsymbol{w})=(\boldsymbol{v},A\boldsymbol{w})\) for \(\boldsymbol{v},\boldsymbol{w}\in H\).

Note that for finite rank operators, this obviously reduces to the standard theorem in linear algebra: a self-adjoint (Hermitian) operator on a finite dimensional space \(V\) with a positive Hermitian form has an orthogonal eigenbasis, and its eigenvalues are real.

36.2 Proof of the Peter-Weyl theorem↩︎

Let \(G\) be a compact Lie group and \(h_N\) a delta-like sequence around \(1\) on \(G\). By replacing \(h_N(x)\) with \(\frac{1}{2}(h_N(x)+h_N(x^{-1}))\), we may assume that \(h_N\) is invariant under inversion. Define the convolution operators \(B_N\) on \(L^2(G)\) by \[(B_N\psi)(y)=\int_G h_N(x)\psi(x^{-1}y)dx=\int_G h_N(yz^{-1})\psi(z)dz.\] By Proposition 229, these operators are compact (as the kernel \(K(y,z):=h_N(yz^{-1})\) is continuous). Moreover, they are clearly self-adjoint (as \(h_N(x)=h_N(x^{-1})\) and \(h_N\) is real) and commute with right translations by \(G\). So by the Hilbert-Schmidt theorem, we have the corresponding spectral decomposition \[L^2(G)={\rm Ker}B_N\oplus \widehat\bigoplus_\lambda H_{N,\lambda}\] invariant under right translations. Since \(H_{N,\lambda}\) are finite dimensional and invariant under right translations, they are contained in \(L^2_{\rm alg}(G)\) (this is the key step of the proof). Thus the closure \(\overline{L^2_{\rm alg}(G)}\) contains the image of \(B_N\). So for any \(\psi\in L^2(G)\) we can find \(\psi_N\in L^2_{\rm alg}(G)\) such that \(||B_N\psi-\psi_N||<\frac{1}{N}\).

Now let \(\psi\in C(G)\). By Cantor’s theorem, \(\psi\) is uniformly continuous. It follows that \(B_N\psi\) uniformly converges to \(\psi\) as \(N\to \infty\) (check it!). Thus \[||\psi-\psi_N||\le ||\psi-B_N\psi||+||B_N\psi-\psi_N||<||\psi-B_N\psi||+\tfrac{1}{N}\to 0\] as \(N\to \infty\). So \(\overline{L^2_{\rm alg}(G)}\) contains \(C(G)\). But \(C(G)\) is dense in \(L^2(G)\) (namely, by using a partition of unity this reduces to the case of a box in \(\Bbb R^n\), where it is well known). Thus \(\overline{L^2_{\rm alg}(G)}=L^2(G)\). This completes the proof of the Peter-Weyl theorem.

36.3 Existence of faithful representations↩︎

36.4 Density in continuous functions↩︎

In fact, we can now prove an even stronger version of the Peter-Weyl theorem. For this note that \(L^2_{\rm alg}(G)\) is a unital algebra.

Let us call a finite dimensional representation \(V\) of a group \(G\) unimodular if \(\wedge^{\dim V}V\cong \Bbb C\) is the trivial representation.

37 Representations of compact topological groups↩︎

37.1 Existence of the Haar measure↩︎

One can generalize integration theory to arbitrary compact and even to locally compact topological groups. For simplicity we will describe this generalization in the case of compact topological groups with a countable base.

Namely, let \(X\) be a compact Hausdorff topological space with a countable base. For compact Hausdorff spaces this is equivalent to being metrizable. Let \(C(X,\Bbb R)\) be the space of continuous real-valued functions on \(X\). This is a real Banach space with norm \[||f||=\max_{x\in X}|f(x)|.\] Recall that by the Riesz-Markov-Kakutani representation theorem, a finite Borel measure \(\mu\) on \(X\) is the same thing as a positive continuous linear functional \(I: C(X,\Bbb R)\to \Bbb R\) (i.e., such that \(I(f)\ge 0\) for \(f\ge 0\)), namely, \[I(f)=\int_X fd\mu.\] Moreover, \(\mu\) is a probability measure if and only if \(I(1)=1\), and any \(\mu\ne 0\) has positive volume and so can be normalized to be a probability measure.

Now let \(G\) be a compact topological group with a countable base. It acts on \(C(G,\Bbb R)\) by left and right translations, so acts on nonnegative probability measures of \(G\).

This measure is also automatically right-invariant (since it is unique) and is called the Haar measure on \(G\).

The proof is the same as for Lie groups, once we have the integration theory, which we now do.

37.2 The Peter-Weyl theorem for compact topological groups↩︎

Again, the proof is analogous to Lie groups, using a delta-like sequence of continuous hat functions. Namely, we may take \[h_N(x)=c_N\max (\tfrac{1}{N}-d(x,1),0),\] where \(d\) is some metric defining the topology of \(G\), and \(c_N>0\) are normalization constants such that \(\int_G h_N(x)dx=1\).

38 The hydrogen atom, I↩︎

38.1 The Schrödinger equation↩︎

Let us now apply our knowledge of non-abelian harmonic analysis to solve a basic problem in quantum mechanics – describe the dynamics of the hydrogen atom.

The mechanics of the hydrogen atom is determined by motion of a charged quantum particle (electron) in a rotationally invariant attracting electric field. The potential of such a field is \(-\frac{1}{r}\), where \(r^2=x^2+y^2+z^2\) (since this theory does not have nontrivial dimensionless quantities, we may choose the units of measurement so that all constants are equal to \(1\)). Thus, the wave function \(\psi(x,y,z,t)\) for our particle obeys the Schrödinger equation \[i\partial_t\psi=H\psi,\] where \(H\) is the quantum Hamiltonian \[H:=-\frac{1}{2}\Delta-\frac{1}{r},\] and \(\Delta=\partial_x^2+\partial_y^2+\partial_z^2\) is the Laplace operator. Recall also that for each \(t\), the function \(\psi(-,-,-,t)\) is in \(L^2(\Bbb R^3)\) and \(||\psi||=1\). The problem is to solve this equation given the initial value \(\psi(x,y,z,0)\).¹⁸

The Schrödinger equation can be solved by separation of variables as follows. Suppose we have an orthonormal basis \(\psi_N\) of \(L^2(\Bbb R^3)\) such that \(H\psi_N=E_N\psi_N\). Then if \[\psi(x,y,z,0)=\sum_N c_N\psi_N(x,y,z)\] (i.e., \(c_N=(\psi,\psi_N)\)) then \[\psi(x,y,z,t)=\sum_N c_Ne^{-iE_Nt}\psi_N(x,y,z),\] So our job is to find such basis \(\psi_N\), i.e., diagonalize the self-adjoint operator \(H\).

Note that the operator \(H\) is unbounded and defined only on a dense subspace of \(L^2(\Bbb R^3)\), and although it is symmetric (\((H\psi,\eta)=(\psi,H\eta)\) for compactly supported functions), it is very nontrivial to say what precisely it means that \(H\) is self-adjoint. Also, this operator turns out to have both discrete and continuous spectrum, which means that there is actually no basis with the desired properties – eigenfunctions of \(H\) which lie in \(L^2(\Bbb R^3)\) span a proper closed subspace of this Hilbert space. However, this will not be a problem for our calculation.

38.2 Bound states↩︎

We first focus on bound states, i.e., solutions of the stationary Schrödinger equation \[H\psi=E\psi\] which belong to \(L^2(\Bbb R^3)\) and thus decay at infinity in the sense of \(L^2\)-norm (this is the situation when the electron does not have enough energy to escape from the nucleus, i.e., it is “bound" to it and thus unlikely to be found far from the origin, which explains the terminology). In particular, such eigenfunctions must have negative energy, \(E<0\). To do so, let us utilize the rotational symmetry and write this equation in spherical coordinates. For this we just need to write the Laplacian \(\Delta\) in spherical coordinates. Let us write \(\boldsymbol{r}=r\boldsymbol{u}\), where \(\boldsymbol{u}\in S^2\) (i.e., \(|\boldsymbol{u}|=1\)). We have \[\Delta=\Delta_r+\tfrac{1}{r^2}\Delta_{\rm sph}\] where \[\Delta_{\rm sph}=\tfrac{1}{\sin^2\phi}\partial_\theta^2+\tfrac{1}{\sin \phi}\partial_\phi \sin\phi \partial_\phi\] is a differential operator on \(S^2\) (the spherical Laplacian, or the Laplace-Beltrami operator) and \[\Delta_r=\partial_r^2+\tfrac{2}{r}\partial_r\] is the radial part of \(\Delta\) (check it!). So our equation looks like \[\partial_r^2\psi+\tfrac{2}{r}\partial_r\psi+\tfrac{2}{r}\psi+\tfrac{1}{r^2}\Delta_{\rm sph}\psi=-2E\psi.\] This equation can be solved by again applying separation of variables. Namely, we look for solutions in the form \[\psi(r,\boldsymbol{u})=f(r)\xi(\boldsymbol{u}),\] where \[\label{sphe} \Delta_{\rm sph} \xi+\lambda \xi=0.\tag{13}\] Then we obtain the following equation for \(f\): \[\label{radialeq} f''(r)+\tfrac{2}{r}f'(r)+(\tfrac{2}{r}-\tfrac{\lambda}{r^2}+2E)f(r)=0.\tag{14}\] So now we have to solve equation 13 and in particular determine which values of \(\lambda\) occur.

To this end, recall that the operator \(\Delta_{\rm sph}\) is rotationally invariant, so it preserves the space \(L^2_{\rm alg}(S^2)\) of functions on \(S^2\) belonging to finite dimensional representations of \(SO(3)\). Moreover, it preserves the decomposition \(L^2_{\rm alg}(S^2)=\oplus_{\ell\ge 0}L_{2\ell}\) of this space into irreducible representations of \(SO(3)\) (Exercise 227(ii)), and on each \(L_{2\ell}\) it acts by a certain scalar \(-\lambda_\ell\). To compute this scalar, consider the vector \(Y_\ell^0\) in \(L_{2\ell}\) of weight zero. This vector is invariant under \(SO(2)\) changing \(\theta\), so it depends only on \(\phi\); in fact, it is a polynomial of degree \(\ell\) in \(\cos\phi\): \(Y_\ell^0=P_\ell(\cos\phi)\). Also orthogonality of the decomposition implies that \[\int_{-1}^1 P_k(z)P_n(z)dz=0,\;k\ne n.\] This means that \(P_n\) are the Legendre polynomials. Also \[\Delta_{\rm sph}P_\ell(z)=\partial_z(1-z^2)\partial_zP_\ell(z)=-\lambda_\ell P_\ell(z),\] which shows (by looking at the leading term) that \[\lambda_\ell=\ell(\ell+1),\;\ell\in \Bbb Z_{\ge 0},\] and the space of solutions of 13 with \(\lambda=\lambda_\ell\) is \(2\ell+1\)-dimensional and is isomorphic to \(L_{2\ell}\) as an \(SO(3)\)-module.

Consider now the vector \(Y_\ell^m\in L_{2\ell}\) of any integer weight \(-\ell\le m\le \ell\). We will be interested in these vectors up to scaling. We have \[Y_\ell^m(\theta,\phi)=e^{im\theta}P_\ell^m(\cos \phi),\] where \(P_{\ell}^m\) are certain functions. These functions are called spherical harmonics. Moreover, it follows from representation theory of \(SO(3)\) that \(Y_\ell^m\) are trigonometric polynomials which are even for even \(m\) and odd for odd \(m\) (check it!), so \(P_\ell^m(z)\) are polynomials in \(z\) when \(m\) is even and are of the form \((1-z^2)^{1/2}\) times a polynomial in \(z\) when \(m\) is odd.

Let us calculate the functions \(P_\ell^m\). Since they are eigenfunctions of the spherical Laplacian, we obtain that \(P_\ell^m\) satisfy the Legendre differential equation \[\partial_z (1-z^2)\partial_zP-\frac{m^2}{1-z^2}P+\ell(\ell+1)P=0.\]

These functions are called associated Legendre polynomials (even though they are not quite polynomials when \(m\) is odd).

Now we can return to equation 14 . It now has the form \[\label{radialeq1} f''(r)+\tfrac{2}{r}f'(r)+(\tfrac{2}{r}-\tfrac{\ell(\ell+1)}{r^2}+2E)f(r)=0.\tag{15}\]

To simplify this equation, write \[f(r)=r^\ell e^{-\frac{r}{n}}h(\tfrac{2r}{n}),\] where \(n\) can be chosen at our convenience. Then for \(h\) we get the equation \[\rho h''(\rho)+(2\ell+2-\rho)h'(\rho)+(n-\ell-1+\tfrac{1}{4}(1+2En^2)\rho)h(\rho)=0.\] We see that the equation simplifies when \(n=\frac{1}{\sqrt{-2E}}\), i.e., \(E=-\frac{1}{2n^2}\), so let us make this choice. Then we have \[\rho h''(\rho)+(2\ell+2-\rho)h'(\rho)+(n-\ell-1)h(\rho)=0,\] which is the generalized Laguerre equation. Moreover, we have \(||\psi||^2<\infty\), which translates to \[\label{integrability} \int_0^\infty \rho^{2\ell+2}e^{-\rho}|h(\rho)|^2d\rho<\infty\tag{16}\] (the factor \(\rho^2\) comes from the Jacobian of the spherical coordinates).

How do solutions of the generalized Laguerre equation behave at \(\rho=0\)? Let us look for a solution of the form \(\rho^s(1+o(1))\). The characteristic equation for \(s\) then has the form \[s(s+2\ell+1)=0,\] which gives \(s=0\) or \(s=-2\ell-1\). Thus, for \(\ell\ge 1\) the solution \(\rho^{-2\ell-1}(1+o(1))\) does not satisfy 16 , so we are left with a unique solution \(h_n(\rho)\) which is regular at \(\rho=0\) and \(h_n(0)=1\). On the other hand, if \(\ell=0\), the solution \(\rho^{-1}(1+o(1))\), even though it satisfies 16 , gives rise to a rotationally invariant function \(\psi\sim \frac{1}{r}\) as \(r\to 0\), so we don’t get \(H\psi=E\psi\), but rather get \(H\psi=E\psi+C\delta_0\), where \(\delta_0\) is the delta function concentrated at zero. So \(\psi\) does not really satisfy the stationary Schrödinger equation as a distribution and has to be discarded, leaving us, as before, with the unique solution \(h_n(\rho)\) such that \(h_n(0)=1\).

Using the power series method, we obtain \[h_n(\rho)=\sum_{k=0}^\infty \frac{(1+\ell-n)...(k+\ell-n)}{(2\ell+2)...(2\ell+1+k)}\frac{\rho^k}{k!}.\] It is easy to see that this series converges for all \(\rho\) and \[\lim_{\rho\to +\infty}\frac{\log h_n(\rho)}{\rho}=1\] unless the series terminates, which happens iff \(n-\ell-1\) is a nonnegative integer. (To check the latter, show that the Taylor coefficients \(a_k\) of \(h_n\) are bounded below by \(\frac{1}{(k+N)!}\) for some \(N\)). So it fails 16 unless \(n-\ell-1\in \Bbb Z_{\ge 0}\). In this case, \[h_n(\rho)=\sum_{k=0}^{n-\ell-1} \frac{(1+\ell-n)...(k+\ell-n)}{(2\ell+2)...(2\ell+1+k)}\frac{\rho^k}{k!}= L^{2\ell+1}_{n-\ell-1}(\rho),\] the \(n-\ell-1\)-th generalized Laguerre polynomial with parameter \(\alpha=2\ell+1\), a polynomial of degree \(n-\ell-1\). Namely, the generalized Laguerre polynomials \(L^\alpha_N\) are defined by the formula \[L_N^\alpha(\rho):=\sum_{k=0}^{N} (-1)^k\frac{N...(N-k+1)}{(\alpha+1)...(\alpha+k)}\frac{\rho^k}{k!}.\] Thus we obtain the following theorem.

39 The hydrogen atom, II↩︎

39.1 Quantum numbers↩︎

The number \(n\) in Theorem 244 is called the principal quantum number; it characterizes the energy of the state. The number \(\ell\) is called the azimuthal quantum number; it characterizes the eigenvalue of the spherical Laplacian \(\Delta_{\rm sph}\), which has the physical interpretation as (minus) the orbital angular momentum operator \(\boldsymbol{L}^2=L_x^2+L_y^2+L_z^2\). By Exercise 245, the operators \(iL_x\), \(iL_y\) and \(iL_z\) are just the generators of the Lie algebra \({\rm Lie}(SO(3))\) acting on \(\Bbb R^3\), i.e., we have \[[L_x,L_y]=-iL_z,\;[L_y,L_z]=-iL_x,\;[L_z,L_x]=-iL_y\] Thus, \(\boldsymbol{L}^2\) is simply a Casimir of \({\rm Lie}(SO(3))\). Namely, recall that the standard Casimir \(C\) acts on \(L_{2\ell}\) as \(\frac{2\ell(2\ell+2)}{4}=\ell(\ell+1)\), so \(\boldsymbol{L}^2=C\).

Finally, \(m\) is called the magnetic quantum number, and it is the eigenvalue of \(L_z=-i\partial_\theta\) (in spherical coordinates).

In fact, this analysis applies not just to hydrogen but to other chemical elements whose nucleus has charge \(>1\), if we neglect interaction between electrons. Thus it can potentially be used to explain patterns of the periodic table.

39.2 Coulomb waves↩︎

We note, however, that \(\psi_{n\ell m}\) do not form a basis of \(L^2(\Bbb R^3)\). Instead, they span (topologically) a proper closed subspace \(L^2_0(\Bbb R^3)\) of \(L^2(\Bbb R^3)\) on which the operator \(H\) is bounded and negative definite. So if a smooth function \(\varphi\) on \(\Bbb R^3\) (say, with compact support away from the origin) satisfies \((H\varphi,\varphi)\ge 0\) then \(\varphi\notin L^2_0(\Bbb R^3)\). It is easy to construct such examples: let \(\varphi\) be a hat function and \(\varphi_s(\boldsymbol{r})=\varphi(\boldsymbol{r}+s\boldsymbol{a})\), where \(\boldsymbol{a}\) is any nonzero vector. We then have \[(H\varphi_s,\varphi_s)=\frac{1}{2}\int_{\Bbb R^3} |\nabla \varphi(\boldsymbol{r})|^2dV-\int_{\Bbb R^3}\frac{|\varphi(\boldsymbol{r})|^2}{|\boldsymbol{r}-s\boldsymbol{a}|}dV,\] and we observe that the first term is positive and the second one goes to zero as \(s\to \infty\), so for large \(s\) this expression is positive. This happens because besides bound states the hydrogen atom also has continuous spectrum \([0,\infty)\) corresponding to free electrons which are not bound by the nucleus. This part of the spectrum can be computed similarly to the discrete (bound state) spectrum, except that the energy will take arbitrary nonnegative values. The corresponding wavefunctions are not normalizable (i.e., not in \(L^2\)), and are given by similar formulas to bound states but with imaginary \(n\). Their continuous linear combinations satisfying appropriate boundary conditions are called Coulomb waves.

39.3 Spin↩︎

Also, the answer \(n^2\) for the number of states in the \(n\)-th energy level does not quite agree with the periodic table, which suggests it should rather be \(2n^2\): the numbers of electrons at each level are \(2,8,18,32...\). This is because the Schrödinger model which we computed is not quite right, as it does not take into account an additional degree of freedom called spin (a sort of intrinsic angular momentum). Namely, it turns out that the space of states of an electron is not \(L^2(\Bbb R^3)\) but rather \(L^2(\Bbb R^3)\otimes \Bbb C^2\), with the same Hamiltonian as before but the Lie algebra \({\rm Lie}(SO(3))\) acting diagonally (where \(\Bbb C^2\) is the 2-dimensional irreducible representation of this Lie algebra). Thus the space of states of the \(n\)-th energy level taking spin into account is \[V_n=(L_0\oplus L_2\oplus...\oplus L_{2n-2})\otimes L_1=2L_1\oplus 2L_3\oplus...\oplus 2L_{2n-3}\oplus L_{2n-1}\] and \(\dim V_n=2n^2\). In other words, we have the additional spin-\(z\) operator \[S_z=\begin{pmatrix} \frac{1}{2}&0\\ 0&-\frac{1}{2} \end{pmatrix}\] acting on the \(\mathbb{C}^2\) factor in the standard basis \(\boldsymbol{e}_+,\boldsymbol{e}_-\). Thus the \(z\)-component of total angular momentum is \[J_z=L_z+S_z,\] and the product states \[\psi_{n\ell m,+}:=\psi_{n\ell m}\otimes \boldsymbol{e}_+,\qquad \psi_{n\ell m,-}:=\psi_{n\ell m}\otimes \boldsymbol{e}_-\] are eigenvectors of \(J_z\) with eigenvalues \(m+\frac{1}{2}\) and \(m-\frac{1}{2}\), respectively.

Note also that \(V_n\) is not a representation of \(SO(3)\) but is only a representation of its double cover \(SU(2)\) where \(-{\rm Id}\) acts by \(-1\). However, this anomaly does not mean a violation of the \(SO(3)\) symmetry, since true quantum states are unit vectors in the Hilbert space up to a phase factor.

39.4 The Pauli exclusion principle↩︎

Suppose now that we have \(k\) electrons, each at the \(n\)-th energy level. If the electrons had been marked, the space of states for them would have been \(V_n^{\otimes k}\). But in real life they are indistinguishable, so we need to mod out by permutations. So we might think the space of states is \(S^kV_n\). However, as electrons are fermions, this answer turns out to be not correct: the correct answer is \(\wedge^k V_n\) rather than \(S^k V_n\). In other words, when two identical electrons are switched, the corresponding vector changes sign. This is another example of a sign which does not violate symmetry since states are well defined only up to a phase factor.

In particular, this implies that if \(k>2n^2\) then the space of states is zero, i.e., there cannot be more than \(2n^2\) electrons at the \(n\)-th energy level (the Pauli exclusion principle). This is exactly the kind of pattern we see in the periodic table.

Namely, the first energy level has two slots (the first row, or period, of the table), and the second one has 8 slots (the second period of the table). Further down interactions between electrons start to matter and the picture is modified (giving still 8 slots in the next period instead of 18), but we still see a similar pattern: 8 slots in the third period, 18 in periods 4,5, and 32 in periods 6,7. This arrangement is justified by the fact that the columns (groups) of elements, which have the same number of electrons at the last level, have similar chemical properties. For example, in the first column we have alkali metals (except hydrogen) and in the last one we have inert gases.

40 Forms of semisimple Lie algebras over an arbitrary field↩︎

40.1 Automorphisms of semisimple Lie algebras↩︎

We showed in Corollary 28 that for a complex semisimple \(\mathfrak{g}\), the group \({\rm Aut}(\mathfrak{g})\) is a Lie group with Lie algebra \(\mathfrak{g}\). We also showed in Theorem 132 that its connected component of the identity \({\rm Aut}(\mathfrak{g})^\circ\) acts transitively on the set of Cartan subalgebras in \(\mathfrak{g}\). This group is called the adjoint group attached to \(\mathfrak{g}\), and we will denote it by \(G_{\rm ad}\).

Let \(\mathfrak{h}\subset \mathfrak{g}\) be a Cartan subalgebra, and \(H\subset G_{\rm ad}\) be the corresponding connected Lie subgroup. This subgroup can be viewed as the group of linear operators \(\mathfrak{g}\to \mathfrak{g}\) which act by \(1\) on \(\mathfrak{h}\) and by \(e^{\alpha(x)}\), \(x\in \mathfrak{h}\), on each \(\mathfrak{g}_\alpha\). Thus the exponential map \(\mathfrak{h}\to H\) defines an isomorphism \(\mathfrak{h}/2\pi iP^\vee\cong H\). The group \(H\) is called the maximal torus of \(G_{\rm ad}\) corresponding to \(\mathfrak{h}\).

In particular, we see that \(H\) is a maximal commutative subgroup of \(G_{\rm ad}\), hence the terminology “maximal torus".

Another obvious subgroup of \({\rm Aut}(\mathfrak{g})\) is the finite group \({\rm Aut}(D)\) of automorphisms of the Dynkin diagram of \(\mathfrak{g}\), which just permutes the generators \(e_i,f_i,h_i\) in the Serre presentation. Thus we have a natural homomorphism \[\xi: {\rm Aut}(D)\ltimes G_{\rm ad}\to {\rm Aut}(\mathfrak{g}),\] which is the identity map on the connected components of \(1\). This homomorphism is clearly injective, since any nontrivial element of \({\rm Aut}(D)\) nontrivially permutes fundamental representations of \(\mathfrak{g}\).

40.2 Forms of semisimple Lie algebras↩︎

We have classified semisimple Lie algebras over \(\Bbb C\), but what about other fields (say of characteristic zero), notably \(\Bbb R\) (the case relevant to the theory of Lie groups)?

To address this question, note that the Serre presentation of a semisimple Lie algebra is defined over \(\Bbb Q\), so it defines a Lie algebra of the same dimension over any such field, by imposing the same generators and relations. Such a Lie algebra is called split. So for example, over an algebraically closed field of characteristic zero, any semisimple Lie algebra is automatically split.

Now let \(\mathfrak{g}\) be a semisimple Lie algebra over a field \(K\) of characteristic zero which splits over a Galois extension \(L\) of \(K\), i.e., \(\mathfrak{g}\otimes_K L=\mathfrak{g}_L\) is split (corresponds to a Dynkin diagram via Serre’s presentation). Can we classify such \(\mathfrak{g}\)?

To this end, let \(\Gamma={\rm Gal}(L/K)\) be the Galois group of \(L\) over \(K\) and observe that we can recover \(\mathfrak{g}\) as the subalgebra of invariants \(\mathfrak{g}_L^\Gamma\). So \(\mathfrak{g}\) is determined by the action of \(\Gamma\) on the split semisimple Lie algebra \(\mathfrak{g}_L\). Note that this action is twisted-linear, i.e., additive and \(g(\lambda x)=g(\lambda)g(x)\) for \(x\in \mathfrak{g}_L\), \(\lambda\in L\), \(g\in \Gamma\). The simplest example of such an action is the action \(\rho_0(g)\) which preserves all the generators \(e_i,f_i,h_i\) and just acts on the scalars, which corresponds to the split form of \(\mathfrak{g}\). So any twisted-linear action \(\rho\) can be written as \[\rho(g)=\eta(g)\rho_0(g)\] for some map \[\eta: \Gamma\to {\rm Aut}(\mathfrak{g}_L).\] In order that \(\rho\) be a homomorphism, we need \[\eta(gh)\rho_0(gh)=\eta(g)\rho_0(g)\eta(h)\rho_0(h),\] which is equivalent to \[\eta(gh)=\eta(g)\cdot g(\eta(h)),\] where for \(a\in {\rm Aut}(\mathfrak{g}_L)\), \(g(a):=\rho_0(g)a\rho_0(g)^{-1}\). In other words, \(\eta\) is a \(1\)-cocycle. We will denote the Lie algebra attached to such cocycle \(\eta\) by \(\mathfrak{g}_\eta\).

It remains to determine when \(\mathfrak{g}_{\eta_1}\) is isomorphic to \(\mathfrak{g}_{\eta_2}\). This will happen exactly when the corresponding representations \(\rho_1\) and \(\rho_2\) are isomorphic, i.e., there is \(a\in {\rm Aut}(\mathfrak{g}_L)\) such that \(\rho_1(g)a=a\rho_2(g)\), i.e., \[\eta_1(g)\rho_0(g)a=a\eta_2(g)\rho_0(g),\] or \[\eta_1(g)=a\eta_2(g)g(a)^{-1}.\] Two 1-cocycles related in this way are called cohomologous (obviously, an equivalence relation), and the set of equivalence classes of cohomologous cocycles is called the first Galois cohomology of \(\Gamma\) with coefficients in \({\rm Aut}(\mathfrak{g}_L)\) and denoted by \(H^1(\Gamma,{\rm Aut}(\mathfrak{g}_L))\). Note that this is cohomology with coefficients in a nonabelian group, so it is just a set and not a group.

So we obtain

40.3 Real forms of a semisimple Lie algebra↩︎

Let us now make this classification more concrete in the case \(K=\Bbb R\), \(L=\Bbb C\), which is relevant to classification of real semisimple Lie groups. In this case, \(\Gamma=\Bbb Z/2\) generated by complex conjugation \(s\mapsto \overline{s}\) and, as we have shown, \({\rm Aut}(\mathfrak{g}_L)={\rm Aut}(D)\ltimes G_{\rm ad}\), where \(D\) is the Dynkin diagram of \(\mathfrak{g}\) and \(G_{\rm ad}\) is the corresponding connected adjoint complex Lie group. Also since we always have \(\eta(1)=1\), the cocycle \(\eta\) is determined by the element \(s=\eta(-1)\in {\rm Aut}(D)\ltimes G_{\rm ad}\). Moreover, \(s\) must satisfy the cocycle condition \[s\overline{s}=1\] and the corresponding real Lie algebra, up to isomorphism, depends only on the cohomology class of \(s\), which is the equivalence class modulo transformations \(s\mapsto as \overline{a}^{-1}\). We thus obtain the following theorem.

We denote the real form of \(\mathfrak{g}\) corresponding to \(s\) by \(\mathfrak{g}_{(s)}\). Namely, \(\mathfrak{g}_{(s)}=\lbrace x\in \mathfrak{g}: \overline{x}=s(x)\rbrace\). For example, \(\mathfrak{g}_{(1)}\) is the split form, consisting of real \(x\in \mathfrak{g}\), i.e., such that \(\overline{x}=x\).

Alternatively, one may define the antilinear involution \(\sigma_s(x)=\overline{s(x)}\), and \(\mathfrak{g}_{(s)}\) is the set of fixed points of \(\sigma_s\) in \(\mathfrak{g}\).

In particular, such \(s\) defines an element \(s_0\in {\rm Aut}(D)\) such that \(s_0^2=1\). Note that the conjugacy class of \(s_0\) is invariant under equivalences. The element \(s_0\) permutes connected components of \(D\), preserving some and matching others into pairs. Thus every semisimple real Lie algebra is a direct sum of simple ones, and each simple one either has a connected Dynkin diagram \(D\) (i.e., the complexified Lie algebra \(\mathfrak{g}\) is still simple) or consists of two identical components (i.e., the complexified Lie algebra is \(\mathfrak{g}=\mathfrak{a}\oplus \mathfrak{a}\) for some simple complex \(\mathfrak{a}\)). In the latter case \(s=(g,\overline{g}^{-1})s_0\) where \(s_0\) is the transposition and \(g\in {\rm Aut}(\mathfrak{a})\), so \(s\) is cohomologous to \(s_0\) by taking \(a=(g,1)\). Thus in this case \(\mathfrak{g}_{(s)}=\mathfrak{g}_{(s_0)}=\mathfrak{a}\), a complex simple Lie algebra regarded as a real Lie algebra.

It remains to consider the case when \(D\) is connected, i.e., \(\mathfrak{g}\) is simple.

So in particular any real form is inner to a unique quasi-split form, and a real form that is both inner and quasi-split is split.

41 Classification of real forms of semisimple Lie algebras↩︎

41.1 The compact real form↩︎

An important example of a real form of simple complex Lie algebra \(\mathfrak{g}\) is the compact real form. It is determined by the automorphism \(\tau\) (called the Cartan involution) defined by the formula \[\tau(h_j)=-h_j,\;\tau(e_j)=-f_j,\;\tau(f_j)=-e_j.\] Let us denote this real form \(\mathfrak{g}_{(\tau)}\) by \(\mathfrak{g}^c\).

Now consider the group \({\rm Aut}(\mathfrak{g}^c)\). Since the Killing form on \(\mathfrak{g}^c\) is negative definite, it is a closed subgroup in the orthogonal group \(O(\mathfrak{g}^c)\), hence is compact. Moreover, it is a Lie group with Lie algebra \(\mathfrak{g}^c\). Thus we obtain

In particular, this gives a new proof that representations of a finite dimensional semisimple Lie algebra are completely reducible (by using Weyl’s unitary trick, see Subsection 35.1). Indeed, let \(G\) be the universal cover of \(G_{\rm ad}\), \(Z\) the center of \(G\), and \(U,V\) two irreducible representations of \(G\) (equivalently, of \(\mathfrak g\)). If \(Z\) acts on \(U,V\) by different characters then obviously \({\rm Ext}^1_{\mathfrak g}(U,V)=0\) (see Subsection 18.1). Otherwise, \({\rm Ext}_{\mathfrak g}^1(U,V)={\rm Ext}^1_{\mathfrak g}(\Bbb C,{\rm Hom}_{\Bbb C}(U,V))\), which is zero since both \(\Bbb C\) and \({\rm Hom}_{\Bbb C}(U,V)\) carry a trivial action of \(Z\) and therefore are representations of \(G_{\rm ad}\), hence of the compact group \(G_{\rm ad}^c\).

41.2 Other examples of real forms↩︎

So let us list real forms of simple Lie algebras that we know so far.

1. Type \(A_{n-1}\). We have the split form \(\mathfrak{sl}_n(\Bbb R)\), the compact form \(\mathfrak{su}(n)\), and also for \(n>2\) the quasi-split form associated to the automorphism \(s(A)=-JA^TJ^{-1}\), where \(J_{ij}=(-1)^i\delta_{i,n+1-j}\) (this automorphism sends \(e_i,f_i,h_i\) to \(e_{n+1-i},f_{n+1-i},h_{n+1-i}\)). So the corresponding real Lie algebra is the Lie algebra of traceless matrices preserving the hermitian or skew-hermitian form defined by the matrix \(J\), which has signature \((p,p)\) if \(n=2p\) and \((p+1,p)\) or \((p,p+1)\) if \(n=2p+1\). Thus in the first case we have \(\mathfrak{su}(p,p)\) and in the second case we have \(\mathfrak{su}(p+1,p)\). Note that for \(n=2\) we have \(\mathfrak{su}(1,1)=\mathfrak{sl}_2(\Bbb R)\), so in this special case this form is not new. We also observe that for \(n\ge 4\) there are other forms, e.g. \(\mathfrak{su}(n-p,p)\) with \(1\le p\le \frac{n}{2}-1\).

2. Type \(B_n\). We have the split form \(\mathfrak{so}(n+1,n)\), the compact form \(\mathfrak{so}(2n+1)\). The Dynkin diagram has no nontrivial automorphisms, so there are no non-split quasi-split forms. In particular, since \(A_1=B_1\), we have \(\mathfrak{so}(3)=\mathfrak{su}(2)\) and \(\mathfrak{so}(2,1)=\mathfrak{su}(1,1)\).

3. Type \(C_n\). We have the split form \(\mathfrak{sp}_{2n}(\Bbb R)\) and compact form \(\mathfrak{u}(n,\Bbb H)\). The Dynkin diagram has no nontrivial automorphisms, so there are no non-split quasi-split forms. The equality \(B_2=C_2\) implies that \(\mathfrak{so}(3,2)=\mathfrak{sp}_4(\Bbb R)\) and \(\mathfrak{so}(5)=\mathfrak{u}(2,\Bbb H)\).

4. Type \(D_n\). We have the split form \(\mathfrak{so}(n,n)\), the compact form \(\mathfrak{so}(2n)\). Moreover, in this case we have a unique nontrivial involution of the Dynkin diagram. More precisely, this is true for \(n\ne 4\), while for \(n=4\) we have \({\rm Aut}(D)=S_3\), but there is still a unique non-trivial involution up to conjugation. So we also have a non-split quasi-split form. To compute it, recall that the split form is defined by the equation \(A=-JA^TJ^{-1}\) where \(J_{ij}=\delta_{i,2n+1-j}\). The quasi-split form is obtained by replacing \(J\) by \(J'=gJ\), where \(g\) permutes \(e_n\) and \(e_{n+1}\) (this is the automorphism that switches \(\alpha_{n-1}\) and \(\alpha_n\) while keeping other simple roots fixed). The signature of the form defined by \(J'\) is \((n+1,n-1)\), so we get that the non-split quasi-split form is \(\mathfrak{so}(n+1,n-1)\). In particular, since \(D_2=A_1+A_1\), for \(n=2\) we get \[\mathfrak{so}(4)=\mathfrak{su}(2)\oplus \mathfrak{su}(2),\;\mathfrak{so}(2,2)=\mathfrak{su}(1,1)\oplus \mathfrak{su}(1,1),\; \mathfrak{so}(3,1)=\mathfrak{sl}_2(\Bbb C)\] (the Lie algebra of the Lorentz group of special relativity). Also, since \(D_3=A_3\), for \(n=3\) we get \(\mathfrak{so}(6)=\mathfrak{su}(4)\), \(\mathfrak{so}(3,3)=\mathfrak{sl}_4(\Bbb R)\), and \(\mathfrak{so}(4,2)=\mathfrak{su}(2,2)\).

5. Type \(G_2\). We have the split and compact forms \(G_2(\Bbb R),G_2^c\).

6. Type \(F_4\). We have the split and compact forms \(F_4(\Bbb R),F_4^c\).

7. Type \(E_6\). We have the split and compact forms \(E_6(\Bbb R),E_6^c\) and the quasi-split form \(E_6^{qs}\) attached to the non-trivial automorphism.

8. Type \(E_7\). We have the split and compact forms \(E_7(\Bbb R),E_7^c\).

9. Type \(E_8\). We have the split and compact forms \(E_8(\Bbb R),E_8^c\).

41.3 Classification of real forms↩︎

However, we are not done with the classification of real forms yet, as we still need to find all real forms and show there are no others. To this end, consider a complex simple Lie algebra \(\mathfrak{g}=\mathfrak{g}^c\otimes_{\Bbb R}\Bbb C\). We have the compact antilinear involution \(\omega=\sigma_\tau\) of \(\mathfrak{g}\) whose set of fixed points is \(\mathfrak{g}^c\). Another real structure on \(\mathfrak{g}\) is then defined by the antilinear involution \(\sigma=\omega\circ g\), where \(g\in {\rm Aut}(\mathfrak{g})\) is such that \({\omega(g)}g=1\). But it is easy to see that \[\omega(g)=(g^\dagger)^{-1},\] where \(x^\dagger\) is the adjoint to \(x\in \mathrm{End}(\mathfrak{g})\) under the negative definite Hermitian form \((X,Y)={\rm Tr}({\rm ad}X{\rm ad}\omega(Y))\) (the Hermitian extension of the Killing form on \(\mathfrak{g}^c\) to \(\mathfrak{g}\)). It follows that the operator \(g\) is self-adjoint. Thus it is diagonalizable with real eigenvalues, and we have a decomposition \[\mathfrak{g}=\oplus_{\gamma\in \Bbb R} \mathfrak{g}(\gamma),\] where \(\mathfrak{g}(\gamma)\) is the \(\gamma\)-eigenspace of \(g\), such that \([\mathfrak{g}(\beta),\mathfrak{g}(\gamma)]=\mathfrak{g}(\beta\gamma)\). Now consider the operator \(|g|^t\) for any \(t\in \Bbb R\). It acts on \(\mathfrak{g}(\gamma)\) by \(|\gamma|^t\), so \(|g|^t=\exp(t\log|g|)\in G_{\rm ad}\) is a 1-parameter subgroup. Now define \(\theta:=g|g|^{-1}\). We have \(\theta\circ \omega=\omega\circ \theta\) and \(\theta^2=1\). Also \(g\) and \(\theta\) define the same real structure since \(\theta=|g|^{-1/2}g\omega(|g|^{1/2})\). This shows that without loss of generality we may assume that \(g=\theta\) with \(\theta\circ\omega=\omega\circ\theta\) (i.e., \(\theta\in \mathop{\mathrm{Aut}}\nolimits(\mathfrak{g}^c)\)) and \(\theta^2=1\).²²

Moreover, another such element \(\theta'\) defines the same real form if and only if \(\theta'=x\theta\omega(x)^{-1}\) for some \(x\in {\rm Aut}(\mathfrak{g})\). So we get \[x\theta\omega(x)^{-1}=\omega(x)\theta x^{-1},\] so setting \(z:=\omega(x)^{-1}x\), we get \(\omega(z)=z^{-1}\), \(\theta z=z^{-1}\theta\). Note that \(z=x^\dagger x\) is positive definite. So setting \(y=xz^{-1/2}\), we have \[\omega(y)=\omega(x)z^{1/2}=xz^{-1/2}=y\] i.e., \(y\in {\rm Aut}(\mathfrak{g}^c)\) and \[\theta'=x\theta\omega(x)^{-1}=x\theta zx^{-1}=xz^{-1/2}\theta z^{1/2}x^{-1}=y\theta y^{-1}.\] Thus we obtain

Theorem 258 provides a different classification of real forms from the one given in Subsection 40.3, obtained by “counting" from the compact form rather than the split form (as we did in Subsection 40.3). We denote the real form of \(\mathfrak{g}\) assigned in Theorem 258 to an involution \(\theta: \mathfrak{g}\to \mathfrak{g}\) by \(\mathfrak{g}_\theta\). For example, \(\mathfrak{g}_1=\mathfrak{g}^c=\mathfrak{g}_{(\tau)}\).

Thus we have a canonical (up to automorphisms of \(\mathfrak{g}^c\)) decomposition \(\mathfrak{g}=\mathfrak{k}\oplus \mathfrak{p}\), into the eigenspaces of \(\theta\) with eigenvalues \(1\) and \(-1\), such that \(\mathfrak{k}\) is a Lie subalgebra, \(\mathfrak{p}\) is a module over \(\mathfrak{k}\) and \([\mathfrak{p},\mathfrak{p}]\subset \mathfrak{k}\). We also have the corresponding decomposition for the underlying real Lie algebra \(\mathfrak{g}^c=\mathfrak{k}^c\oplus \mathfrak{p}^c\). Moreover, the corresponding real form \(\mathfrak{g}_\theta\) is just \(\mathfrak{g}_\theta=\mathfrak{k}^c\oplus \mathfrak{p}_\theta\), where \(\mathfrak{p}_\theta:=i\mathfrak{p}^c\).

Thus we have a decomposition \(\mathfrak{h}=\mathfrak{h}_+\oplus \mathfrak{h}_-\), and \(\theta\) acts by \(1\) on \(\mathfrak{h}_+\) and by \(-1\) on \(\mathfrak{h}_-\).

By Lemma 45, a generic element \(t\in \mathfrak{h}_+\) is regular in \(\mathfrak{g}\). So let us pick one for which \({\rm Re}(t,\alpha^\vee)\) is nonzero for any coroot \(\alpha^\vee\) of \(\mathfrak{g}\), and use it to define a polarization of \(R\): set \(R_+:=\lbrace\alpha\in R:{\rm Re}(t,\alpha^\vee)>0\rbrace\). Then \(\theta(R_+)=R_+\). So \(\theta(\alpha_i)=\alpha_{\theta(i)}\), where \(\theta(i)\) is the action of \(\theta\) on the Dynkin diagram \(D\) of \(\mathfrak{g}\). Thus if \(\theta(i)=i\) then \(\theta(e_i)=\pm e_i\), \(\theta(h_i)=h_i\), \(\theta(f_i)=\pm f_i\) while if \(\theta(i)\ne i\), we can normalize \(e_i,e_{\theta(i)},f_i,f_{\theta(i)}\) so that \(\theta(e_i)=e_{\theta(i)}\), \(\theta(f_i)=f_{\theta(i)}\), \(\theta(h_i)=h_{\theta(i)}\). Thus \(\theta\) can be encoded in a marked Dynkin diagram of \(\mathfrak{g}\): we connect vertices \(i\) and \(\theta(i)\) if \(\theta(i)\ne i\) and paint a \(\theta\)-stable vertex \(i\) white if \(\theta(e_i)=e_i\) (i.e., \(e_i\in \mathfrak{k}\), a compact root), and black if \(\theta(e_i)=-e_i\) (i.e., \(e_i\in \mathfrak{p}\), a non-compact root). Such a decorated Dynkin diagram is called a Vogan diagram. So we see that every Vogan diagram gives rise to a real form, and every real form is defined by some Vogan diagram.

41.4 Real forms of classical Lie algebras↩︎

We are not finished yet with the classification of real forms since different Vogan diagrams can define the same real form (they could arise from different choices of \(R_+\) coming from different choices of the element \(t\)). However, we are now ready to classify real forms of classical Lie algebras.

1. Type \(A_{n-1}\), compact inner class. In this case \(\theta\) is an inner automorphism, conjugation by an element of order \(\le 2\) in \(PSU(n)\). Obviously, such an element can be lifted to \(g\in U(n)\) such that \(g^2=1\), so \(\theta(x)=gxg^{-1}\). Thus \(g={\rm Id}_p\oplus (-{\rm Id}_q)\) where \(p+q=n\) and we may assume that \(p\ge q\). It is easy to see that this defines the real form \(\mathfrak{g}_\theta=\mathfrak{su}(p,q)\), and \(\mathfrak{k}=\mathfrak{sl}_p\oplus \mathfrak{gl}_q\). These are all pairwise non-isomorphic since the corresponding automorphisms \(\theta\) are not conjugate to each other. So we get \([\frac{n}{2}]+1\) real forms. Note that for \(n=2\) this exhausts all real forms, so we have only two – \(\mathfrak{su}(2)\) and \(\mathfrak{su}(1,1)=\mathfrak{sl}_2(\Bbb R)\) with \(\mathfrak{k}=\mathfrak{gl}_1\).

2. Type \(A_{n-1}\), \(n>2\), the split inner class. If \(n\) is odd, there is no choice as all the vertices of the Vogan diagram are connected into pairs, so we only get the split form \(\mathfrak{g}_\theta=\mathfrak{sl}_n(\Bbb R)\). However, if \(n=2k\) is even, there is one unmatched vertex in the middle of the Vogan diagram, which can be either white or black. It is easy to check that in the first case (white vertex) \(\mathfrak{k}=\mathfrak{sp}_{2k}\) and in the second one (black vertex) \(\mathfrak{k}=\mathfrak{so}_{2k}\). So the first case is \(\mathfrak{g}_\theta=\mathfrak{sl}(k,\Bbb H)\), the Lie algebra of quaternionic matrices of size \(k\) whose trace has zero real part (See Subsection 6.3), while the second case is the split form \(\mathfrak{g}_\theta=\mathfrak{sl}_n(\Bbb R)\).

3. Type \(B_n\). Then \(\theta\) is an inner automorphism, given by an element of order \(\le 2\) in \(SO(2n+1)\). So \(\theta={\rm Id}_{2p+1}\oplus (-{\rm Id}_{2q})\) where \(p+q=n\). Thus all the real forms are \(\mathfrak{so}(2p+1,2q)\) (all distinct), \(\mathfrak{k}=\mathfrak{so}_{2p+1}\oplus \mathfrak{so}_{2q}\).

4. Type \(C_n\). Then \(\theta\) is an inner automorphism, given by an element \(g\in {\rm Sp}_{2n}(\Bbb C)\) such that \(g^2=1\) or \(g^2=-1\). In the first case the \(1\)-eigenspace of \(g\) has dimension \(2p\) and the \(-1\)-eigenspace has dimension \(2q\) (since they are symplectic), where \(p+q=n\), and we may assume \(p\ge q\) (replacing \(g\) by \(-g\) if needed). So the real form we get is \(\mathfrak{g}_\theta=\mathfrak{u}(p,q,\Bbb H)\), the quaternionic pseudo-unitary Lie algebra for a quaternionic Hermitian form (see Subsection 6.3). In this case \(\mathfrak{k}=\mathfrak{sp}_{2p}\oplus \mathfrak{sp}_{2q}\). On the other hand, if \(g^2=-1\) then \(\Bbb C^{2n}=V(i)\oplus V(-i)\) (eigenspaces of \(g\), which in this case are Lagrangian subspaces), so \(\mathfrak{k}=\mathfrak{gl}_n(\Bbb C)\). The corresponding real form is the split form \(\mathfrak{g}_\theta=\mathfrak{sp}_{2n}(\Bbb R)\).

5. Type \(D_n\), compact inner class. We again have an inner automorphism \(\theta\) given by \(g\in SO(2n)\) such that \(g^2=\pm 1\). If \(g^2=1\) then \(\Bbb C^{2n}=V(1)\oplus V(-1)\), the direct sum of eigenspaces, and since \(\det(g)=1\), the eigenspaces are even-dimensional, of dimensions \(2p\) and \(2q\) where \(p+q=n\), and, as in the case of type \(C_n\), we may assume \(p\ge q\). So the corresponding real form is \(\mathfrak{g}_\theta=\mathfrak{so}(2p,2q)\) with \(\mathfrak{k}=\mathfrak{so}_{2p}\oplus \mathfrak{so}_{2q}\). On the other hand, if \(g^2=-1\) then we have \(\Bbb C^{2n}=V(i)\oplus V(-i)\), and these are Lagrangian subspaces of dimension \(n\). So \(\mathfrak{k}=\mathfrak{gl}_n(\Bbb C)\). The corresponding real form is the quaternionic orthogonal Lie algebra (symmetries of a quaternionic skew-Hermitian form), \(\mathfrak{g}_\theta=\mathfrak{so}^*(2n)\) (see Subsection 6.3).

6. Type \(D_n\), the other inner class. In this case \(\theta\) is given by an element \(g\) of \(O(2n)\) such that \(\det(g)=-1\) and \(g^2=\pm 1\). Note that if \(g^2=-1\) then , as shown above, \(\det(g)=1\), so in the case at hand we always have \(g^2=1\). Then \(\Bbb C^{2n}=V(1)\oplus V(-1)\), but now the dimensions of these spaces are odd, \(2p+1\) and \(2q+1\) where \(p+q=n-1\), and we may assume that \(p\ge q\). So the real form is \(\mathfrak{g}_\theta=\mathfrak{so}(2p+1,2q+1)\), with \(\mathfrak{k}=\mathfrak{so}_{2p+1}\oplus\mathfrak{so}_{2q+1}\). Note that for \(n=3\), \(D_3=A_3\), so we have \(\mathfrak{so}(5,1)=\mathfrak{sl}(2,\Bbb H)\). Note also that this agrees with what we found before: the split form \(\mathfrak{so}(n,n)\) is in the compact inner class for even \(n\) and in the other one for odd \(n\), and the quasi-split form \(\mathfrak{so}(n+1,n-1)\) the other way around.

42 Real forms of exceptional Lie algebras↩︎

42.1 Equivalence of Vogan diagrams↩︎

For exceptional Lie algebras, it is convenient to make a more systematic use of Vogan diagrams (we could do this also for classical Lie algebras, but there we can also do everything explicitly using linear algebra). Recall that any real form comes from a certain Vogan diagram, but different Vogan diagrams may be equivalent, i.e., define the same real form. So our job is to describe this equivalence relation.

First consider the case of the compact inner class. In this case the Vogan diagram is just the Dynkin diagram with black and white vertices (i.e., no matched vertices). Moreover, the case of all white vertices corresponds to the compact form, while the case when there are black vertices to noncompact forms. So let us focus on the latter case. Thus we have an element \(\theta\in H\subset G_{\rm ad}\) such that \(\theta\ne 1\) but \(\theta^2=1\), but we are allowed to conjugate \(\theta\) by elements of \(N(H)\), i.e., transform it by elements of the Weyl group \(W\). So how do simple reflections \(s_i\) act on \(\theta\) (in terms of its Vogan diagram)?

The Vogan diagram of \(\theta\) is determined by the numbers \(\alpha_j(\theta)=\pm 1\): if this number is \(1\) then \(j\) is white, and if it is \(-1\) then \(j\) is black. Now, we have \[\alpha_j(s_i(\theta))=(s_i\alpha_j)(\theta)=(\alpha_j-a_{ij}\alpha_i)(\theta)=\alpha_j(\theta)\alpha_i(\theta)^{-a_{ij}}.\] This equals \(\alpha_j(\theta)\) unless \(\alpha_i(\theta)=-1\) and \(a_{ij}\) is odd. Thus we obtain the following lemma.

The same lemma holds, with the same proof, in the case of the other inner class (which for exceptional Lie algebras is possible only for \(E_6\)), except we should ignore the vertices matched into pairs (so \(i\) and \(j\) should be \(\theta\)-stable vertices).

42.2 Classification of real forms↩︎

We are now ready to classify real forms of exceptional Lie algebras.

1. Type \(G_2\). We have two color configurations up to equivalence: \(\circ\circ\) and \((\bullet\circ,\circ\bullet,\bullet\bullet)\). The first corresponds to the compact form \(G_2^c\) and the second to the split form \(G_2^{\rm spl}\). It is easy to check that in the second case \(\mathfrak{k}=\mathfrak{sl}_2\oplus \mathfrak{sl}_2\) (indeed, it has dimension \(6\) and rank \(2\)). So we don’t have other real forms.

2. Type \(F_4\). Let \(\alpha_1,\alpha_2\) be short roots and \(\alpha_3,\alpha_4\) long roots. Then all nonzero off-diagonal \(a_{ij}\) are odd except \(a_{23}=-2\). So we may change the colors of the neighbors of any black vertex, except that if the black vertex is \(2\) then we should not change the color of \(3\). By such changes, we can bring the colors at \(3,4\) into the form \(\circ\circ\) or \(\circ\bullet\), and then bring the colors at \(1,2\) to the form \(\circ\circ\) or \(\bullet\circ\). So we are down to four configurations: \[\circ\circ\circ\circ,\;\bullet\circ\circ\circ,\;\circ\circ\circ\bullet,\;\bullet\circ\circ\bullet\]

Moreover, the fourth case, \(\bullet\circ\circ\bullet\), is actually equivalent to the third one, \(\circ\circ\circ\bullet\). This is seen from the chain of equivalences \[\circ\circ\circ\bullet=\circ\circ\bullet\bullet=\circ\bullet\bullet\circ= \bullet\circ\bullet\circ=\bullet\bullet\bullet\bullet=\bullet\bullet\circ\bullet=\bullet\circ\circ\bullet\] Thus we are left with three variants, \[\circ\circ\circ\circ, \bullet\circ\circ\circ, \circ\circ\circ\bullet.\] The first configuration, \(\circ\circ\circ\circ\), corresponds to the compact form \(F_4^c\).

In the second case, \(\bullet\circ\circ\circ\), \(\alpha(\theta)=-1\) exactly when the root \(\alpha\) has half-integer coordinates (recall that there are 16 such roots, see Subsection 23.3). Thus the Lie algebra \(\mathfrak{k}\) is comprised by the root subspaces for roots with integer coordinates and the Cartan subalgebra, i.e., \(\mathfrak{k}=\mathfrak{so}_9\) (type \(B_4\)). Also in this case \(\mathfrak{p}=S\), the spin representation of \(\mathfrak{so}_9\). This is not the split form, since for the split form \(\dim \mathfrak{k}\) should be \(24\) and here it is \(36\). Let us denote this form \(F_4^1\).

Thus, the third case, \(\circ\circ\circ\bullet\), must be the split form, \(F_4^{\rm spl}\). We see that \(\mathfrak{k}\) contains the 21-dimensional Lie algebra \(\mathfrak{sp}_6=C_3\) (generated by the simple roots \(\alpha_1,\alpha_2,\alpha_3\)), so given that \(\mathfrak{k}\) has rank \(4\) and dimension \(24\), we have \(\mathfrak{k}=\mathfrak{sp}_6\oplus \mathfrak{sl}_2\).

3. Type \(E_6\), split inner class. In this case in the Vogan diagram two pairs of vertices are connected, so we can only color the two remaining vertices. So we have two equivalence classes of colorings – \(\circ\circ\) and \((\bullet\bullet,\bullet\circ,\circ\bullet)\). Let us show that they correspond to two different real forms. Consider first the \(\circ\circ\) case. In this case \(\theta\) is simply the diagram automorphism, so we have \(\mathfrak{k}=F_4\), as the Dynkin diagram of \(F_4\) is obtained by folding the Dynkin diagram of \(E_6\) (check it!). This is not the split form since \(\dim \mathfrak{k}=52\), but for the split form it is \(36\); denote this form by \(E_6^1\). So the split form \(E_6^{\rm spl}\) corresponds to the second equivalence class \((\bullet\bullet,\bullet\circ,\circ\bullet)\). One can show that in this case \(\mathfrak{k}=\mathfrak{sp}_8\), i.e., type \(C_4\) (check it!).

4. \(E_6,E_7,E_8\), compact inner class. In this case the Vogan diagram has no arrows and just is the usual Dynkin diagram with vertices colored black and white. One option is that all vertices are white, this corresponds to the compact forms \(E_6^c, E_7^c,E_8^c\) (\(\theta=1\)). If there is at least one black vertex, then by using equivalence transformations we can make sure that the nodal vertex is black. Then flipping the color of its neighbors if needed, we can make sure that the vertex on the shortest leg is also black. This allows us to change the color of the nodal vertex whenever we want (as long as the vertex on the shortest leg remains black).

We now want to unify the coloring of the long leg. We can bring the long leg to the following normal forms:

\(E_6\): \(\circ\circ, \bullet\circ=\bullet\bullet=\circ\bullet\). But by flipping the colors on the neighbors of the nodal vertex, we see that \(\bullet\circ\) and \(\circ\circ\) are equivalent, so all patterns are equivalent to \(\bullet\bullet\).

\(E_7\): \(\circ\circ\circ, \bullet\circ\circ=\bullet\bullet\circ=\circ\bullet\bullet=\circ\circ\bullet\), \(\bullet\circ\bullet=\bullet\bullet\bullet=\circ\bullet\circ\). But by flipping the colors on the neighbors of the nodal vertex, we see that all patterns are equivalent to \(\bullet\bullet\bullet\).

\(E_8\): \(\circ\circ\circ\circ, \bullet\circ\circ\circ=\bullet\bullet\circ\circ=\circ\bullet\bullet\circ=\circ\circ\bullet\bullet=\circ\circ\circ\bullet,\) \[\bullet\circ\circ\bullet=\bullet\circ\bullet\bullet=\bullet\bullet\bullet\circ=\begin{cases}\bullet\circ\bullet\circ\\ \circ\bullet\circ\circ\end{cases}\] \[=\bullet\bullet\bullet\bullet=\bullet\bullet\circ\bullet= \circ\bullet\bullet\bullet=\begin{cases}\circ\bullet\circ\bullet \\ \circ\circ\bullet\circ\end{cases}.\] But by flipping the colors on the neighbors of the nodal vertex, we see that all patterns are equivalent to \(\bullet\bullet\bullet\bullet\).

Thus we can always arrange all vertices on the long leg except possibly the neighbor of the node to be black, while the short leg and the node also remain black. In addition, as seen from the pictures above, in the cases \(E_6\) and \(E_8\) these two configurations are equivalent by transformations inside the leg.

Now we can consider the configurations on the remaining leg (of length 2). The equivalence classes are \(\circ\circ\) and \(\bullet\circ=\circ\bullet=\bullet\bullet\).

So in the case of \(E_6\) and \(E_8\) we get just two cases. It turns out that both for \(E_6\) and \(E_8\) these give two different real forms, one of which is split in the case of \(E_8\).

Consider first the \(E_6\) case. One option is to take the Vogan diagram with just one black vertex, at the end of the long leg: \[\boxed{\begin{matrix}\circ & \circ & \circ &\circ &\bullet\\ & & \circ & & &\end{matrix}}\] Then \(\mathfrak{k}=\mathfrak{so}_{10}\oplus \mathfrak{so}_2\) (as the black vertex corresponds to a minuscule weight). We denote this real form by \(E_6^2\). On the other hand, if there is only one black vertex on the short leg, \[\boxed{\begin{matrix}\circ & \circ & \circ &\circ &\circ \\ & & \bullet & & &\end{matrix}}\] then \(\mathfrak{k}\) contains \(\mathfrak{sl}_6\), so this real form is different (as \(\mathfrak{sl}_6\) is not a Lie subalgebra of \(\mathfrak{so}_{10}\)). It’s not difficult to show that in this case \(\mathfrak{k}=\mathfrak{sl}_6\oplus \mathfrak{sl}_2\). We denote this real form by \(E_6^3\).

Now consider the \(E_8\) case. Again one option is the Vogan diagram with just one black vertex, at the end of the long leg: \[\boxed{\begin{matrix}\circ & \circ & \circ &\circ &\circ &\circ &\bullet\\ & & \circ & & &\end{matrix}}\] Then \(\mathfrak{k}\) contains \(E_7\), so this is not the split form since \(\dim \mathfrak{k}\ge 133\) but for the split form it should be 120. In fact, it is not hard to see that \(\mathfrak{k}=E_7\oplus \mathfrak{sl}_2\). We denote this real form by \(E_8^1\). The second form is the split one, \(E_8^{\rm spl}\). It can, for example, be obtained if we color black only one vertex, at the end of the middle leg: \[\boxed{\begin{matrix}\bullet & \circ & \circ &\circ &\circ &\circ &\circ \\ & & \circ & & &\end{matrix}}\] In fact, it’s not hard to show that the algebra \(\mathfrak{k}\) in this case is \(\mathfrak{so}_{16}\).

Finally, consider the \(E_7\) case. In this case we have four options, but two of them end up being equivalent. Namely, we have \[\boxed{\begin{matrix}\bullet & \circ & \bullet &\circ &\bullet&\bullet\\ & & \bullet & & &\end{matrix}}= \boxed{\begin{matrix}\bullet & \bullet & \bullet &\bullet &\bullet&\bullet\\ & & \circ & & &\end{matrix}}= \boxed{\begin{matrix}\circ & \bullet & \circ &\bullet &\bullet&\bullet\\ & & \circ & & &\end{matrix}}=\] \[\boxed{\begin{matrix}\circ & \bullet & \bullet &\bullet &\circ&\bullet\\ & & \circ & & &\end{matrix}}= \boxed{\begin{matrix}\circ & \circ & \bullet &\circ &\circ&\bullet\\ & & \bullet & & &\end{matrix}}= \boxed{\begin{matrix}\circ & \circ & \bullet &\circ &\bullet&\bullet\\ & & \bullet & & &\end{matrix}}.\] So we are left with three cases, which all turn out different. The first one is just one black vertex at the end of the long leg: \[\boxed{\begin{matrix}\circ & \circ & \circ &\circ &\circ &\bullet\\ & & \circ & & &\end{matrix}}\] In this case \(\mathfrak{k}\) contains \(E_6\), so this is not the split form, as \(\dim \mathfrak{k}\ge 78\) but for the split form it is \(63\). It is easy to see that \(\mathfrak{k}=E_6\oplus \mathfrak{so}_2\) in this case (the black vertex corresponds to the minuscule weight). We denote this real form by \(E_7^1\). The second option is a black vertex at the end of the middle leg: \[\boxed{\begin{matrix}\bullet & \circ & \circ &\circ &\circ &\circ \\ & & \circ & & &\end{matrix}}\] Then \(\mathfrak{k}\) contains \(\mathfrak{so}_{12}\), of dimension \(66\), so again not the split form. One can show that for this form \(\mathfrak{k}=\mathfrak{so}_{12}\oplus \mathfrak{sl}_2\). We denote it by \(E_7^2\). Finally, the split form \(E_7^{\rm spl}\) is obtained when one colors black just the end of the short leg: \[\boxed{\begin{matrix}\circ & \circ & \circ &\circ &\circ &\circ \\ & & \bullet & & &\end{matrix}}\] Then \(\mathfrak{k}\) contains \(\mathfrak{sl}_7\) and one can show that \(\mathfrak{k}=\mathfrak{sl}_8\).

43 Classification of connected compact and complex reductive groups↩︎

43.1 Connected compact Lie groups↩︎

We are now ready to classify connected compact Lie groups. We start with the following exercise.

Now we can proceed. We already know many examples of compact connected Lie groups - namely tori \((S^1)^r\) and also groups \(G_{\rm ad}^c\) where \(G_{\rm ad}={\rm Aut}(\mathfrak{g})^\circ\) for a semisimple Lie algebra \(\mathfrak{g}\). We can also consider products \((S^1)^r\times G_{\rm ad}^c\). Exercise 268 shows that the Lie algebra of any compact Lie group is isomorphic to one of such a product, so this should be an exhaustive list up to taking coverings and quotients by finite central subgroups. It thus remains to understand the nature of these coverings, which reduces to understanding \(\pi_1(G_{\rm ad}^c)\). So our next task is to compute this group. In particular, we will show that it is finite.

So let \(\mathfrak{g}\) be a semisimple complex Lie algebra and \(G\) the corresponding simply connected complex Lie group (the universal cover of \(G_{\rm ad}\)). Let \(Z\) be the kernel of the covering map \(G\to G_{\rm ad}\), which is also \(\pi_1(G_{\rm ad})\) and the center of \(G\). The finite dimensional representations of \(G\) are the same as those of \(\mathfrak{g}\), so the irreducible ones are \(L_\lambda\), \(\lambda\in P_+\). The center \(Z\) acts by a certain character \({\boldsymbol{\chi}}_\lambda: Z\to \Bbb C^\times\) on each \(L_\lambda\). Since \(L_{\lambda+\mu}\) is contained in \(L_\lambda\otimes L_\mu\), we have \({\boldsymbol{\chi}}_{\lambda+\mu}={\boldsymbol{\chi}}_\lambda{\boldsymbol{\chi}}_\mu\), so \({\boldsymbol{\chi}}\) uniquely extends to a homomorphism \({\boldsymbol{\chi}}: P\to \mathrm{Hom}(Z,\Bbb C^\times)\). Also, by definition \({\boldsymbol{\chi}}_\theta=1\) (since the maximal root \(\theta\) is the highest weight of the adjoint representation on which \(Z\) acts trivially).

Now, by Exercise 193, if \(\lambda(h_i)\) are sufficiently large then for every root \(\alpha\) of \(\mathfrak{g}\) we have \(L_{\lambda+\alpha}\subset L_\lambda\otimes \mathfrak{g}\). Thus \({\boldsymbol{\chi}}_{\lambda+\alpha}={\boldsymbol{\chi}}_\lambda\), hence \({\boldsymbol{\chi}}_\alpha=1\). So \({\boldsymbol{\chi}}\) is trivial on the root lattice \(Q\), i.e., defines a homomorphism \(P/Q\to \mathrm{Hom}(Z,\Bbb C^\times)\), or, equivalently, \(Z\to P^\vee/Q^\vee\).

Note that the same argument works for \(G_{\rm ad}^c\), its universal cover \(G^c\), and its center \(Z^c\) instead of \(G_{\rm ad}\), \(G\), \(Z\).

Now we can proceed with the classification of semisimple compact connected Lie groups. We begin with the following lemma from topology (see e.g. [10], Supplementary exercises to Chapter 13, p.500, Exercise 4).

This immediately implies the following corollary.

In particular, it follows that simply connected semisimple compact Lie groups are of the form \(\prod_{i=1}^nG_i^c\), where \(G_i^c\) are simply connected and simple.²³

43.2 Polar decomposition↩︎

Now let us study the structure of the Lie subgroup \(G_{{\rm ad},\theta}\subset G_{\rm ad}\) corresponding to the real form \(\mathfrak{g}_\theta\subset \mathfrak{g}\) of a semisimple complex Lie algebra \(\mathfrak{g}\), namely, the group of fixed points of the antiholomorphic involution \(\omega_\theta=\omega\circ \theta\) in \(G_{\rm ad}\). It is clear that this subgroup is closed (\({\rm Lie}\overline{G_{{\rm ad},\theta}}=\mathfrak{g}_\theta={\rm Lie}G_{{\rm ad},\theta}\)), but it may be disconnected: e.g. if \(\mathfrak{g}_\theta=\mathfrak{sl}_2(\Bbb R)\) then \(G_{\rm ad}=PGL_2(\Bbb C)\), so \(G_{{\rm ad},\theta}=PGL_2(\Bbb R)\), the quotient of \(GL_2(\Bbb R)\) by scalars, which has two components. However, the results below apply mutatis mutandis to the connected group \(G_{{\rm ad},\theta}^\circ\).

Let \(K^c\subset G_{{\rm ad},\theta}\) be the subgroup of elements acting on \(\mathfrak{g}\) by unitary operators; namely, \(K^c\) is the set of fixed points of \(\omega_\theta\) on \(G_{\rm ad}^c\).²⁴ This is a closed (possibly disconnected) subgroup of \(G_{\rm ad}^c\) since \({\rm Lie}\overline{K^c}=\mathfrak{k}^c={\rm Lie}K^c\), hence it is compact. Also let \(P_\theta:=\exp(\mathfrak{p}_\theta)\subset G_{{\rm ad},\theta}\) (note that it is not a subgroup!). Since \(\mathfrak{p}_\theta\) acts on \(\mathfrak{g}\) by Hermitian operators, the exponential map \(\exp: \mathfrak{p}_\theta\to P_\theta\) is a diffeomorphism, so \(P_\theta\subset G_{{\rm ad},\theta}\) is a closed embedded submanifold (the set of elements acting on \(\mathfrak{g}\) by positive Hermitian operators).

In particular, applying Theorem 271 to complex Lie groups, we get

This also implies that by taking coverings the polar decomposition applies verbatim to the real form \(G_\theta=G^{\omega_\theta}\subset G\) of any connected complex semisimple Lie group \(G\) instead of \(G_{\rm ad}\). We note, however, that if \(G\) is simply connected, then \(G_\theta^\circ\) need not be. In fact, its fundamental group could be infinite. The simplest example is \(G=SL_2(\Bbb C)\), then for the split form \(G_\theta=SL_2(\Bbb R)\), which as we showed is homotopy equivalent to \(SO(2)=S^1\), i.e. its fundamental group is \(\Bbb Z\).

43.3 Connected complex reductive groups↩︎

It is clear that the Lie algebra \({\rm Lie}G\) of any complex reductive Lie group \(G\) is reductive, and any complex reductive Lie algebra is the Lie algebra of a connected complex reductive Lie group. However, a simply connected complex Lie group with a reductive Lie algebra need not be reductive (e.g. \(G=\Bbb C\)).

If \(G=((\Bbb C^\times)^m\times G_{\rm ss})/Z\) is a connected complex reductive Lie group then by Corollary 64, \(Z\subset (S^1)^m\times G_{\rm ss}^c\subset (\Bbb C^\times)^m\times G_{\rm ss}\), so we can define the compact subgroup \(G^c\subset G\) by \(G^c:=((S^1)^m\times G_{\rm ss}^c)/Z\), and vice versa. It is easy to see that this gives rise to a complexification functor \(K\mapsto K_{\Bbb C}\) \[\lbrace\text{connected compact Lie groups}\rbrace \to \lbrace\text{connected reductive complex Lie groups}\rbrace\] such that \((G^c)_{\Bbb C}=G\), which defines a bijection between isomorphism classes. Moreover, restriction of finite dimensional representations from \(G\) to \(G^c\) is an equivalence, so representations of \(G\) are completely reducible. The irreducible representations are parametrized by collections \((n_1,...,n_m,\lambda)\), \(\lambda\in P_+(G_{\rm ss})\), \(n_i\in \Bbb Z\), which define the trivial character of \(Z\).

We also see that the preimage \(H\subset G\) of the standard maximal torus \(H_{\rm ad}\subset G_{\rm ad}\) (see Subsection 40.1) is a torus \(H=\exp(\mathfrak{h})\cong (\Bbb C^\times)^r\subset G\) which contains the center \(Z\) of \(G\), and if \(G\) is semisimple simply connected then the exponential map defines an isomorphism \(\mathfrak{h}/2\pi i Q^\vee\cong H\).

43.4 Linear groups↩︎

A connected Lie group \(G\) (real or complex) is called linear if it can be realized as a Lie subgroup of \(GL_n(\Bbb R)\), respectively \(GL_n(\Bbb C)\). We have seen that any complex semisimple group is linear. However, for real semisimple groups this is not so (e.g. the universal cover of \(SL_2(\Bbb R)\) is not linear, see Exercise 72). In fact, we see that we can characterize connected real semisimple linear groups as follows.

43.5 Strongly regular elements in connected complex reductive groups↩︎

In this subsection we would like to generalize the results of Subsections 20.1 and 20.2 on strongly regular elements from Lie algebras to Lie groups. We repeat the arguments from these subsections with suitable changes.

Let \(G\) be a connected complex reductive group of rank \(r\).

Now let \(P_g(t)\) be the characteristic polynomial of the operator \(\mathop{\mathrm{Ad}}\nolimits_g\) on \(\mathfrak{g}\). By Lemma 48, \(P_g(t+1)\) is divisible by \(t^r\) for all \(g\in G\). Thus \[\label{ptp1} P_g(t+1)=t^r(t^m+b_{m-1}(g)t^{m-1}+\cdots+b_0(g)),\tag{18}\] where \(b_i\) are regular functions on \(G\), in fact polynomials in the matrix coefficients of the adjoint representation, and \(m=\dim\mathfrak{g}-r\).

44 Maximal tori in compact groups, Cartan decomposition↩︎

44.1 Maximal tori in connected compact Lie groups↩︎

Let \(\mathfrak{g}\) be a complex semisimple Lie algebra, \(\mathfrak{g}^c\) its compact form, \(G\) a connected Lie group with Lie algebra \(\mathfrak{g}\), \(G^c\subset G\) its compact part (the connected Lie subgroup with Lie algebra \(\mathfrak{g}^c\)), as above.

A Cartan subalgebra \(\mathfrak{h}^c\subset \mathfrak{g}^c\) is a maximal commutative Lie subalgebra (note that it automatically consists of semisimple elements since all elements of \(\mathfrak{g}^c\) are semisimple). In other words, it is a subspace such that \(\mathfrak{h}^c\otimes_{\Bbb R}\Bbb C\) is a Cartan subalgebra of \(\mathfrak{g}\).

Recall that all Cartan subalgebras of \(\mathfrak{g}\) are conjugate, even if equipped with a system of simple roots (Theorem 132). Namely, given two such subalgebras \((\mathfrak{h},\Pi)\) and \((\mathfrak{h}',\Pi')\), there is \(g\in G\) such that \({\rm Ad}_g (\mathfrak{h},\Pi)=(\mathfrak{h}',\Pi')\). It turns out that the same result holds for \(\mathfrak{g}^c\).

Note that for every Cartan subalgebra \(\mathfrak{h}^c\subset \mathfrak{g}^c\), \(H^c=\exp(\mathfrak{h}^c)\subset G^c\) is a torus, which is clearly a maximal torus: the Lie algebra of any torus \(T\subset G^c\) containing \(H^c\) would be a commutative subalgebra of \(\mathfrak g^c\) containing \(\mathfrak{h}^c\), so it must coincide with \(\mathfrak h^c\), hence \(T=H^c\). Conversely, if \(H^c\subset G^c\) is a maximal torus then \({\rm Lie}(H^c)\) can be included in a Cartan subalgebra, hence it is itself a Cartan subalgebra. So the exponential map defines a bijection between Cartan subalgebras in \(\mathfrak{g}^c\) and maximal tori in \(G^c\). The same statements apply to Cartan subalgebras in \(\mathfrak{g}\) and maximal tori in \(G\). Moreover, by the last paragraph of Subsection 43.3, \(H^c,H\) are maximal abelian subgroups of \(G^c,G\), since the same is true for \(G_{\rm ad}\) (Subsection 40.1).

We also have

This immediately implies

44.2 Semisimple and unipotent elements↩︎

Let \(G\) be a connected reductive complex Lie group. An element \(g\in G\) is called semisimple if it acts in every finite dimensional representation of \(G\) by a semisimple (=diagonalizable) operator, and unipotent if it acts in every finite dimensional representation of \(G\) by a unipotent operator (all eigenvalues are \(1\)). For example, Corollary 65(ii) implies that every regular semisimple element is semisimple.

Note that the same argument applied to \(G^c,\mathfrak{g}^c\) instead of \(G,\mathfrak{g}\) gives another proof of Theorem 277(ii).

In view of Proposition 284, strongly regular elements are also called regular semisimple.

44.3 Maximal abelian subspaces of \(\mathfrak{p}_\theta\)↩︎

Let \(G\) be a connected complex semisimple group, \(G_\theta\subset G\) a real form, \(\mathfrak{g}_\theta\subset \mathfrak{g}\) their Lie algebras. We have the polar decomposition \(G_\theta=K^c P_\theta\) and the additive version \(\mathfrak{g}_\theta=\mathfrak{k}^c\oplus \mathfrak{p}_\theta\), with \(\mathfrak{p}_\theta=i\mathfrak{p}^c\). Also \(\mathfrak{g}^c=\mathfrak{k}^c\oplus \mathfrak{p}^c\).

44.4 The Cartan decomposition of semisimple linear groups↩︎

Let \(\mathfrak{a}\subset \mathfrak{p}_\theta\) be a maximal abelian subspace and \(A=\exp(\mathfrak{a})\subset P_\theta \subset G_\theta\). This is a subgroup isomorphic to \(\Bbb R^n\), where \(n=\dim \mathfrak{a}\).

44.5 Maximal compact subgroups↩︎

44.6 Cartan subalgebras in real semisimple Lie algebras↩︎

Recall that a Cartan subalgebra of a real semisimple Lie algebra \(\mathfrak{g}_\theta\) is a maximal commutative subalgebra consisting of semisimple elements. We have seen that Cartan subalgebras in a complex semisimple Lie algebra are conjugate, but this is not so for real semisimple Lie algebras, as demonstrated by the following exercise.

Let us say that a semisimple element of \(\mathfrak{g}_\theta\) is split if it acts on \(\mathfrak{g}_\theta\) with real eigenvalues, and say that a commutative Lie subalgebra of \(\mathfrak{g}_\theta\) is a split subalgebra if it consists of split elements. An invariant of a Cartan subalgebra \(\mathfrak{h}\subset \mathfrak{g}_\theta\) under conjugation is the dimension \(s(\mathfrak{h})\) of the largest split subalgebra of \(\mathfrak{h}\) (consisting of all split elements of \(\mathfrak{h}\)). For example, a split real form \(\mathfrak{g}_\theta\) has a split Cartan subalgebra with \(s(\mathfrak{h})=r={\rm rank}\mathfrak{g}\), and conversely, a real form that admits a split Cartan subalgebra is split. Also, in Exercise 290, \(s(\mathfrak{h}_m)=n-1-m\).

Let us say that \(\mathfrak{h}\) is maximally split if \(s(\mathfrak{h})\) is the largest possible, and maximally compact if \(s(\mathfrak{h})\) is the smallest possible. For example, in Exercise 290, \(\mathfrak{h}_0\) is maximally split and \(\mathfrak{h}_{[n/2]}\) is maximally compact (where \([n/2]\) is the floor of \(n/2\)). Also, a split Cartan subalgebra is maximally split and a compact one (i.e., one for which \(\exp(\mathfrak{h})\) is a compact torus) is maximally compact, if they exist. Finally, the Cartan subalgebra \(\mathfrak{h}_+^c\oplus i\mathfrak{h}_-^c\), where \(\mathfrak{h}_+^c,\mathfrak{h}_-^c\) are as in the proof of Proposition 260, is maximally compact.

Note that \(s(\mathfrak{h})\) may also be interpreted as the signature of the Killing form restricted to \(\mathfrak{h}\), which equals \((s(\mathfrak{h}),r-s(\mathfrak{h}))\).

44.7 Integral form of the Weyl character formula↩︎

Thus we see that the orthogonality of characters can be written as \[\frac{1}{|W|}\int_T \chi_\lambda(t)\overline{\chi_\mu(t)}|\Delta(t)|^2dt=\delta_{\lambda,\mu}.\]

45 Topology of Lie groups and homogeneous spaces, I↩︎

45.1 The Chevalley-Eilenberg complex of a compact connected Lie group↩︎

We would now like to study topology of connected Lie groups. The Cartan decomposition implies that any real semisimple Lie group \(G_\theta\) is diffeomorphic to the product of its maximal compact subgroup \(K^c\) and a Euclidean space. This combined with weak Levi decomposition (Theorem 104) implies that topology of connected Lie groups essentially reduces to topology of compact ones, as any simply-connected solvable Lie group has a filtration by normal subgroups with successive quotients being the 1-dimensional group \(\Bbb R\), hence is diffeomorphic to \(\Bbb R^n\) (cf. Theorem 321, Corollary 78 below).

So let us study cohomology of compact connected Lie groups.

We first recall some generalities on cohomology of manifolds. As we mentioned before, the cohomology of an \(n\)-dimensional manifold \(M\) can be computed by the de Rham complex \[0\to \Omega^0(M)\to \Omega^1(M)\to...\to \Omega^n(M)\to 0,\] where \(\Omega^i(M)\) is the space of smooth (complex-valued) differential \(i\)-forms on \(M\). The maps in this complex are given by the differential \(d: \Omega^i(M)\to \Omega^{i+1}(M)\), which satisfies the equation \(d^2=0\). Namely, we define the \(i\)-th de Rham cohomology of \(M\) as the quotient \[H^i(M,\Bbb C):=\Omega^i_{\rm closed}(M)/\Omega^i_{\rm exact}(M)\] where \(\Omega^i_{\rm closed}(M)\subset \Omega^i(M)\) is the space of closed forms (such that \(d\omega=0\)) and \(\Omega^i_{\rm exact}(M)\subset \Omega^i(M)\) is the space of exact forms (such that \(\omega=d\eta\) for some \(\eta\in \Omega^{i-1}(M)\)).

If \(M\) is compact then the spaces \(H^i(M,\Bbb C)\) are known to be finite dimensional, so we can define the Betti numbers of \(M\), \(b_i(M):=\dim H^i(M,\Bbb C)\). Note that \(b_0(M)\) is the number of connected components of \(M\), so if \(M\) is connected then \(b_0(M)=1\).

The wedge product of differential forms descends to the cohomology, which makes \(H^{\bullet}(M,\Bbb C):=\oplus_{i=0}^n H^i(M,\Bbb C)\) into a graded algebra. This algebra is associative and graded-commutative: \(ab=(-1)^{\deg(a)\deg(b)}ba\) (since the wedge product of differential forms has these properties). Moreover, if \(f: M\to N\) is a differentiable map of manifolds then we have the pullback map \(f^*:\Omega^i(N)\to \Omega^i(M)\) which commutes with \(d\) and hence descends to the cohomology. Also \(f^*\) preserves the wedge product, hence defines a graded algebra homomorphism \(f^*: H^{\bullet}(N,\Bbb C)\to H^{\bullet}(M,\Bbb C)\).

Recall that for a vector field \(v\) on \(M\), the Lie derivative \[L_v: \Omega^\bullet(M)\to \Omega^\bullet(M)\] is the unique derivation of the algebra of differential forms which commutes with the de Rham differential and equals the usual derivative of a function along \(v\) on \(\Omega^0(M)\).

Suppose now that a compact connected Lie group \(G\) acts on a manifold \(M\). Then we have the averaging operator \(P: \Omega^{\bullet}(M)\to \Omega^{\bullet}(M)\) over \(G\) which commutes with \(d\) and satisfies the equation \(P^2=P\), so we have a decomposition of complexes \[\Omega^{\bullet}(M)=\Omega^{\bullet}(M)^G\oplus \Omega^{\bullet}(M)_0\] where the first summand is the image of \(P\) and the second one is the kernel of \(P\).

The complex \(\Omega^{\bullet}(G)^G\) is called the Chevalley-Eilenberg complex of \(G\).

45.2 Cohomology of Lie algebras↩︎

It turns out that the Chevalley-Eilenberg complex of \(G\) can be described purely algebraically in terms of the Lie algebra \(\mathfrak{g}={\rm Lie}(G)_{\Bbb C}\). To this end, we will need another lemma from basic differential geometry.

Now observe that \(\Omega^m(G)^G=\wedge^m\mathfrak{g}^*\). Thus we get

This purely algebraic complex can be defined for any Lie algebra \(\mathfrak{g}\) over any field (the equality \(d^2=0\) follows from the Jacobi identity).²⁸ It is called the standard complex or the Chevalley-Eilenberg complex of \(\mathfrak{g}\), denoted \(CE^{\bullet}(\mathfrak{g})\), and its cohomology is called the Lie algebra cohomology of \(\mathfrak{g}\), denoted \(H^{\bullet}(\mathfrak{g})\).²⁹

Also note that the complex \(CE^{\bullet}(\mathfrak{g})\) has wedge product multiplication, which descends to the cohomology. Thus \(H^{\bullet}(\mathfrak{g})\) is a graded-commutative associative algebra. Furthermore, if \(\mathfrak{g}={\rm Lie}(G)_{\Bbb C}\) for a compact connected Lie group \(G\) then \(H^{\bullet}(\mathfrak{g})\cong H^{\bullet}(G,\Bbb C)\) as a graded algebra. However, this may fail even at the level of vector spaces (i.e., Betti numbers) if \(G\) is not compact.

We also have

Thus it suffices to determine the cohomology of simple, simply connected compact Lie groups.

46 Topology of Lie groups and homogeneous spaces, II↩︎

46.1 The coproduct on the cohomology ring↩︎

To understand the algebra \(R:=H^{\bullet}(G)=H^{\bullet}(G,\Bbb C)\) better, note that the multiplication map \(G\times G\to G\) induces the graded algebra homomorphism \(\Delta: H^{\bullet}(G)\to H^{\bullet}(G\times G)=H^{\bullet}(G)\otimes H^{\bullet}(G)\), which is coassociative: \[(\Delta\otimes \text{id})\circ \Delta=(\text{id}\otimes \Delta)\circ \Delta.\] (Note that the warning in Remark 296 about tensor product in the graded sense still applies here!) Such a map \(\Delta\) is called a coproduct since it defines an algebra structure on the dual space \(R^*\) (see Subsection 12.3). We also have the augmentation map \(\varepsilon: R\to \Bbb C\) such that \[(\varepsilon\otimes 1)(\Delta(x))=(1\otimes \varepsilon)(\Delta(x))=x\] for all \(x\in R\). Such a structure is called a graded bialgebra.³⁰

Let us now determine the number \(r\). We have \(2^r=\dim (\wedge^{\bullet}\mathfrak{g}^*)^\mathfrak{g}\). But this dimension can be computed using the Weyl character formula. Namely, the character of \(\wedge^{\bullet} \mathfrak{g}^*\) is \[\chi_{\wedge^{\bullet}\mathfrak{g}^*}(t)=2^{{\rm rank}\mathfrak{g}}\prod_{\alpha>0}(1+\alpha(t))(1+\alpha(t)^{-1}),\] where \(T\subset G\) is a maximal torus and \(t\in T\). So \[\dim (\wedge^{\bullet}\mathfrak{g}^*)^\mathfrak{g}=\frac{2^{{\rm rank}\mathfrak{g}}}{|W|}\int_T\prod_{\alpha>0}(\alpha(t^2)-1)(1-\alpha(t^{-2}))dt =2^{{\rm rank}\mathfrak{g}}.\] So \(r={\rm rank}\mathfrak{g}\).

Thus we have \[H^{\bullet}(G)=H^{\bullet}(\mathfrak{g})=(\wedge^{\bullet} \mathfrak{g}^*)^\mathfrak{g}=\wedge^{\bullet} (\xi^{(1)},...,\xi^{(r)}),\] where \(r={\rm rank}\mathfrak{g}\) and \(\deg(\xi^{(i)})=2m_i+1\). Moreover, it suffices to consider the case when \(\mathfrak{g}\) is simple. What are the numbers \(m_i\) in this case?

Let us order \(m_i\) as follows: \(m_1\le m_2\le...\le m_r\). We know that \(r+2\sum m_i=\dim \mathfrak{g}\), so \(\sum_i m_i=|R_+|\). Also it is not hard to see that \(m_1=1\), \(m_2>1\):

In particular, we see that for a simple compact connected Lie group \(G\), one has \(H^3(G,\Bbb C)\cong \Bbb C\). Thus, the sphere \(S^n\) admits a Lie group structure if and only if \(n=0,1,3\).

46.2 The cohomology ring of a simple compact connected Lie group↩︎

In fact, we have the following classical theorem, which we will not prove in general, but will prove below for type \(A\) and also in exercises for classical groups and \(G_2\).

A modern general proof of this theorem can be found in [12].

We will prove Theorem 301 in the case of type \(A\).

Thus we get that the cohomology \(H^{\bullet}(U(n),\Bbb C)=H^{\bullet}(GL_n(\Bbb C),\Bbb C)\) is \(\wedge^{\bullet}(\xi_1,\xi_3,...,\xi_{2n-1})\) (where subscripts are degrees) with Poincaré polynomial 23 , and \(H^{\bullet}(SU(n),\Bbb C)=H^{\bullet}(SL_n(\Bbb C),\Bbb C)= \wedge^{\bullet}(\xi_3,...,\xi_{2n-1})\) with Poincaré polynomial \((1+q^3)(1+q^5)...(1+q^{2n-1})\).

In the next exercise and the following subsections we will use the notions of a cell complex and its cellular homology and cohomology with coefficients in any commutative ring, and the fact that if a manifold is equipped with a cell decomposition (i.e., represented as a disjoint union of cells) then its cellular cohomology with \(\Bbb C\)-coefficients (=dual to the cellular homology) is canonically isomorphic to the de Rham cohomology via the integration pairing (the de Rham theorem). More details can be found, for instance, in [13].

46.3 Cohomology of homogeneous spaces↩︎

Let \(G\) be a connected compact Lie group, \(\mathfrak{g}={\rm Lie}(G)_{\Bbb C}\), \(K\subset G\) a closed subgroup, \(\mathfrak{k}={\rm Lie}(K)_\Bbb C\), and consider the homogeneous space \(G/K\). How to compute the cohomology \(H^{\bullet}(G/K,\Bbb C)\)?

Since the group \(G\) acts on \(G/K\), this cohomology is computed by the complex \(\Omega^{\bullet}(G/K)^G=(\wedge^{\bullet} (\mathfrak{g}/\mathfrak{k})^*)^K\). Let us denote this complex by \(CE^{\bullet}(\mathfrak{g},K)\). It is called the relative Chevalley-Eilenberg complex.

For example, if \(K=\Gamma\) is finite, this is just the \(\Gamma\)-invariant part of the usual Chevalley-Eilenberg complex. But \(\Gamma\) acts trivially on the cohomology, so we get \(H^{\bullet}(G/\Gamma)=H^{\bullet}(G)\) (as already noted above).

But what happens if \(\dim K>0\)? Can we describe the differential in this complex algebraically as we did for \(K=1\)?

This question is answered by the following proposition. Let \(\mathfrak{k}\subset \mathfrak{g}\) be a pair of Lie algebras (not necessarily finite dimensional, over any field). Denote by \(CE^i(\mathfrak{g},\mathfrak{k})\) the spaces \((\wedge^i(\mathfrak{g}/\mathfrak{k})^*)^\mathfrak{k}\).

Now note that, going back to the setting of compact Lie groups, we have \(CE^\bullet(\mathfrak{g},K)=CE^\bullet(\mathfrak{g},\mathfrak{k})^{K/K^\circ}\), so we obtain

Thus, the computation of the cohomology of \(G/K\) reduces to the computation of the relative Lie algebra cohomology, which is again a purely algebraic problem.

47 Topology of Lie groups and homogeneous spaces, III↩︎

47.1 Grassmannians↩︎

Let \(G=U(m+n), K=U(n)\times U(m)\), so that \(G/K\) is the Grassmannian \({\rm G}_{m+n,n}(\Bbb C)\cong {\rm G}_{m+n,m}(\Bbb C)\) (the manifold of \(m\)-dimensional or \(n\)-dimensional subspaces of \(\Bbb C^{m+n}\)). The element \(s=I_n\oplus (-I_m)\) acts by \(-1\) on \(\mathfrak{g}/\mathfrak{k}=V\otimes W^*\oplus W\otimes V^*\), where \(V,W\) are the tautological representations of \(U(n)\) and \(U(m)\). So we get that the Grassmannian has cohomology only in even degrees, and \[H^{2i}({\rm G}_{m+n,m}(\Bbb C))=\wedge^{2i}(V\otimes W^*\oplus W\otimes V^*)^{U(n)\times U(m)}.\] We can therefore use the skew Howe duality (Proposition 189) to see that \[\dim H^{2i}({\rm G}_{m+n,m}(\Bbb C))=N_i(n,m),\] where \(N_i(n,m)\) is the number of partitions \(\lambda=(\lambda_1,...,\lambda_k)\) whose Young diagram has \(i\) boxes and fit into the rectangle \(m\times n\) (i.e., such that \(k\le m,\lambda_1\le n\)).

To compute \(N_i(m,n)\), consider the generating function \[f_{n,m}(q)=\sum_i N_i(n,m)q^{i}.\] Then, denoting by \(p_i\) the jumps \(\lambda_i-\lambda_{i+1}\) of \(\lambda\) (with \(p_0=n-\lambda_1\)), we have \[\sum_{n\ge 0}f_{n,m}(q)z^n=\] \[\sum_{p_0,p_1,...,p_m\ge 0} z^{p_0+p_1+...+p_m}q^{p_1+2p_2+...+mp_m}=\prod_{j=0}^m \frac{1}{1-q^jz}.\] So the Betti numbers of Grassmannians are the coefficients of this series. For example, if \(m=1\) we get \[\sum_{n\ge 0}f_{n,m}(q)z^n=\frac{1}{(1-z)(1-qz)}=\sum_n (1+q+...+q^n)z^n.\] So we recover the Poincaré polynomial \(1+q+...+q^n\) of the complex projective space \(\Bbb C\Bbb P^n\). More precisely, this is the Poincaré polynomial evaluated at \(q^{\frac{1}{2}}\), which is actually a polynomial in \(q\) since we have nontrivial cohomology only in even degrees.

The polynomials \(f_{n,m}(q)\) are called the Gaussian binomial coefficients and they can be computed explicitly. Namely, we have

47.2 Schubert cells↩︎

There is actually a more geometric way to obtain the same result. This way is based on decomposing the Grassmannians into Schubert cells. Namely, let \(F_i\subset \Bbb C^{m+n}\) be spanned by the first \(i\) basis vectors \(e_1,...,e_i\); thus \[0=F_0\subset F_1\subset...\subset F_{m+n}=\Bbb C^{m+n}.\] Given an \(m\)-dimensional subspace \(V\subset \Bbb C^{m+n}\), let \(\ell_j\) be the smallest integer for which \(\dim (F_{\ell_j}\cap V)=j\). Then \[1\le \ell_1<\ell_2<...<\ell_m\le m+n,\] which defines a partition with parts \[\lambda_1=\ell_m-m, \lambda_2=\ell_{m-1}-m+1,...,\lambda_m=\ell_1-1\] fitting in the \(m\times n\) box. Let \(S_\lambda\subset {\rm G}_{m+n,m}(\Bbb C)\) be the set of \(V\) giving such numbers \(\lambda_i\).

So we see that \({\rm G}_{m+n,m}(\Bbb C)\) has a cell decomposition into a disjoint union of Schubert cells.

Now we can rederive the same formula for the Poincaré polynomial of the Grassmannian from the following well-known fact from algebraic topology:

Indeed, the boundary map in this cell complex has to be zero, and its fundamental group must be trivial, as it is a quotient of the fundamental group of the 1-skeleton of \(X\), which is a single point (why?).

So we obtain an even stronger statement than before:

In particular, this gives Betti numbers over any field (including positive characteristic), not just \(\Bbb C\).

47.3 Flag manifolds↩︎

The flag manifold \(\mathcal{F}_n(\Bbb C)\) is the space of all complete flags \(0=V_0\subset V_1\subset...\subset V_n=\Bbb C^n\), where \(\dim V_i=i\). Note that the flag manifold is a homogeneous space: \({\mathcal{F}}_n=G/T\), where \(G=U(n)\) and \(T=U(1)^n\) is a maximal torus in \(G\). It can also be written as \(G_{\Bbb C}/B\), where \(G_{\Bbb C}=GL_n(\Bbb C)\) and \(B=B_n\) is the subgroup of upper triangular matrices.

We have fibrations \(\pi: \mathcal{F}_n(\Bbb C)\to \Bbb C\Bbb P^{n-1}\) sending \((V_1,...,V_{n-1})\) to \(V_{n-1}\), whose fiber is the space of flags in \(V_{n-1}\), i.e., \(\mathcal{F}_{n-1}(\Bbb C)\). This shows, by induction, that flag manifolds can be decomposed into even-dimensional cells isomorphic to \(\Bbb C^k\).

More precisely, to define actual cells, we need to trivialize the fibration \(\pi\) over each cell in \(\Bbb C\Bbb P^{n-1}\). These cells are \(C_{in}\), \(i=1,...,n\), where \(C_{in}\) is the set of hyperplanes \(E\subset \Bbb C^n\) defined by an equation \(a_1x_1+...+a_nx_n=0\) where the first nonzero coefficient is \(a_i\) (so \(C_{in}\cong \Bbb C^{n-i}\)). This means that for \((x_1,...,x_n)\in E\), the coordinates \(x_j,j\ne i\) can be chosen arbitrarily, and then \(x_i\) is uniquely determined. So we may identify \(E\) with \(\Bbb C^{n-1}\) by sending \((x_1,...,x_n)\) to \((x_1,...,x_{i-1},x_{i+1},...,x_n)\), which defines the required trivialization.

Thus we obtain a stratification of \(\mathcal{F}_n\) into cells \(C_w\) labeled by permutations \(w\in S_n\), which we’ll represent as orderings of \(1,2,...,n\). Namely, this stratification and labeling are defined by induction in \(n\): for \(w\in S_{n-1}\), \(C_w\times C_{in}=C_{w_i'}\), where \(w_i'\in S_n\) is obtained from \(w\) by inserting \(n\) in the \(i\)-th place (namely, \(w_i'=w\circ (i,i+1,...,n))\). By analogy with the Grassmannian, the cells \(C_w\) are called Schubert cells.

It follows that the Betti numbers of \(\mathcal{F}_n\) vanish in odd degrees, and in even degrees are given by the generating function \[\sum b_{2i}(\mathcal{F}_n)q^i=[n]_q!=(1+q)(1+q+q^2)...(1+q+...+q^{n-1}).\] Moreover, it is easy to see that \(\dim_{\Bbb C} C_w=\ell(w)\), so we get the identity \[\sum_{w\in S_n}q^{\ell(w)}=[n]_q!\] Finally, note that the group \(B_n\) of upper triangular matrices preserves each \(C_w\). In fact, it is easy to check by induction in \(n\) that \(C_w\) are simply \(B_n\)-orbits on \({\mathcal{F}}_n\).

We can also define the partial flag manifold \(\mathcal{F}_S(\Bbb C)\), where \(S\subset [1,n-1]\) is a subset, namely the space of partial flags \((V_s, s\in S)\), \(V_s\subset \Bbb C^n\), \(\dim V_s=s\), \(V_s\subset V_t\) if \(s<t\).

48 Levi decomposition↩︎

48.1 Cohomology of Lie algebras with coefficients↩︎

The definition of cohomology of Lie algebras may be generalized to define the cohomology with coefficients in a module, so that the cohomology considered above is the one for the trivial module.

Let \(\mathfrak{g}\) be a Lie algebra and \(V\) a \(\mathfrak{g}\)-module. The Chevalley-Eilenberg (or standard) complex of \(\mathfrak{g}\) with coefficients in \(V\) is defined by \[CE^\bullet(\mathfrak{g},V):=\mathrm{Hom}(\wedge^\bullet\mathfrak{g},V)\] with differential defined by the full Cartan formula (without dropping the first term): \[d\omega(a_0,...,a_m)=\sum_i (-1)^i a_i\omega(a_0,...,\widehat a_i,...,a_m)+\] \[\sum_{i<j}(-1)^{i+j}\omega([a_i,a_j],a_0,...,\widehat a_i,...,\widehat{a_j},...,a_m).\] The cohomology of this complex is called the cohomology of \(\mathfrak{g}\) with coefficients in \(V\) and denoted \(H^\bullet(\mathfrak{g},V)\). Note that the previously defined cohomology \(H^\bullet(\mathfrak{g})\) is \(H^\bullet(\mathfrak{g},\Bbb C)\).

If \(\mathfrak{g}\) is the Lie algebra of a Lie group \(G\) (or its complexification) and \(V\) is finite dimensional, then we simply have \(CE^\bullet(\mathfrak{g},V):=(\Omega^\bullet(G)\otimes V)^G\) (and the differential is just the de Rham differential). So in particular by Theorem 295 we have (using that if \(\mathfrak{g}\) is semisimple then the smallest \(i>0\) such that \(H^i(\mathfrak{g},\Bbb C)\ne 0\) is \(3\)):

However, this cohomology is non-trivial in general if \(\mathfrak{g}\) is not semisimple or \(V\) is infinite dimensional.

Let us explore the meaning of \(H^i(\mathfrak{g},V)\) for small \(i\).

1. We have \(H^0(\mathfrak{g},V)=V^\mathfrak{g}\), the \(\mathfrak{g}\)-invariants in \(V\).

2. \(H^1(\mathfrak{g},V)\) is the quotient of the space \(Z^1(\mathfrak{g},V)\) of 1-cocycles
\(\omega: \mathfrak{g}\to V\), i.e., linear maps satisfying \[\omega([x,y])=x\omega(y)-y\omega(x)\] by the space of 1-coboundaries \(B^1(\mathfrak{g},V)\), of the form \(\omega(x)=xv\) for some \(v\in V\).

3. \(Z^1(\mathfrak{g},\mathfrak{g})\) is the Lie algebra of derivations of \(\mathfrak{g}\), and \(B^1(\mathfrak{g},\mathfrak{g})\) is the ideal of inner derivations. So \(H^1(\mathfrak{g},\mathfrak{g})\) is the Lie algebra of outer derivations, the quotient of all derivations by inner derivations. In particular, we rederive the fact proved earlier that all derivations of a semisimple complex Lie algebra \(\mathfrak{g}\) are inner (\(H^1(\mathfrak{g},\mathfrak{g})=0\)).

4. Suppose we want to define an abelian extension \(\widetilde{\mathfrak{g}}\) of \(\mathfrak{g}\) by \(V\), i.e., a Lie algebra which can be included in the short exact sequence \[0\to V\to \widetilde{\mathfrak{g}}\to \mathfrak{g}\to 0\] where \(V\) is an abelian ideal. To classify such extensions, pick a vector space splitting \(\widetilde{\mathfrak{g}}=\mathfrak{g}\oplus V\), then the commutator looks like \[[(x,v),(y,w)]=([x,y],xw-yv+\omega(x,y)),\] where \(\omega: \wedge^2\mathfrak{g}\to V\) is a linear map. The Jacobi identity is then equivalent to \(\omega\) being in the space \(Z^2(\mathfrak{g},V)\) of 2-cocycles. Moreover, it is easy to check that for two such extensions \(\widetilde{\mathfrak{g}}_1,\widetilde{\mathfrak{g}}_2\) there is an isomorphism \(\phi: \widetilde{\mathfrak{g}}_1\to \widetilde{\mathfrak{g}}_2\) which acts trivially on \(V\) and \(\mathfrak{g}\) if and only if the corresponding cocycles \(\omega_1,\omega_2\) differ by a coboundary: \(\omega_1-\omega_2\in B^2(\mathfrak{g},V)\). Thus, we get

5. Let us now study deformations of Lie algebras. Suppose \(\mathfrak{g}\) is a Lie algebra over a field \(\boldsymbol{k}\) and we want to deform the bracket, with deformation parameter \(t\). So the new bracket will be \[[x,y]_t=[x,y]+tc_1(x,y)+t^2c_2(x,y)+...,\] where \(c_i: \wedge^2\mathfrak{g}\to \mathfrak{g}\) are linear maps. This bracket should satisfy the Jacobi identity, i.e., define a new Lie algebra structure on \(\mathfrak{g}[[t]]\) (over \(\boldsymbol{k}[[t]]\)). Such deformations are distinguished up to linear isomorphisms \[a=1+ta_1+t^2a_2+...\] where \(a_i\in \mathrm{End}_{\boldsymbol{k}}(\mathfrak{g})\).

In particular, in first order, i.e., modulo \(t^2\), we get a new Lie algebra structure on \(\mathfrak{g}[t]/t^2\mathfrak{g}[t]=\mathfrak{g}\oplus t\mathfrak{g}\) such that this Lie algebra can be included in the short exact sequence \[0\to t\mathfrak{g}\to \mathfrak{g}\oplus t\mathfrak{g}\to \mathfrak{g}\to 0\] where \(t\mathfrak{g}\cong \mathfrak{g}\) is an abelian ideal with adjoint action of \(\mathfrak{g}\) (note that this Lie algebra structure is automatically \(\boldsymbol{k}[t]/t^2\)-linear). So this is an abelian extension of \(\mathfrak{g}\) by \(t\mathfrak{g}\), and we know that such extensions are classified by \(H^2(\mathfrak{g},\mathfrak{g})\). So we obtain

Thus if \(H^2(\mathfrak{g},\mathfrak{g})=0\), every deformation is isomorphic to the trivial one, with \(c_1=c_2=...=0\). Indeed, applying automorphisms
\(a=1+ta_1+t^2a_2+...\), we can kill successively \(c_1\), then \(c_2\), then \(c_3\), and so on. Thus from Whitehead’s theorem we obtain

However, not all first order deformations of a Lie algebra lift to second order, i.e., modulo \(t^3\). Namely, the Jacobi identity in the second order tells us that \(dc_2=[c_1,c_1]\), where \([c_1,c_1]\) is the Schouten bracket of \(c_1\) with itself: \[[c_1,c_1](x,y,z)=c_1(c_1(x,y),z)+c_1(c_1(y,z),x)+c_1(c_1(z,x),y).\] This expression is automatically a cocycle (check it!), but we need it to be a coboundary. So the cohomology class of \([c_1,c_1]\) in \(H^3(\mathfrak{g},\mathfrak{g})\) is an obstruction to lifting the deformation modulo \(t^3\). Thus the space \(H^3(\mathfrak{g},\mathfrak{g})\) is the home for obstructions to deformations. For example, if \(\mathfrak{g}\) is abelian then \(H^2(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}_{\boldsymbol{k}}(\wedge^2\mathfrak{g},\mathfrak{g})\), and the obstruction to extending \(c=tc_1\) modulo \(t^3\) is \[{\rm Jacobi}(c_1):=[c_1,c_1]\in H^3(\mathfrak{g},\mathfrak{g})=\mathrm{Hom}_{\boldsymbol{k}}(\wedge^3\mathfrak{g},\mathfrak{g}).\]

6. In a similar way we can study deformations \(V[[t]]\) of a module \(V\) over \(\mathfrak{g}\): \[\rho_t(x)=\rho(x)+t\rho_1(x)+t^2\rho_2(x)+...\] Modulo \(t^2\) we get a \(\mathfrak{g}\)-module structure on \(V[t]/t^2V[t]=V\oplus tV\) such that we have a short exact sequence \[0\to tV\to V\oplus tV\to V\to 0.\] Thus first order deformations of \(V\) are classified by \({\rm Ext}^1_\mathfrak{g}(V,V)=H^1(\mathfrak{g},\mathrm{End}_{\boldsymbol{k}}V)\). Again, lifting of this deformation modulo \(t^3\) is not automatic, and we get an obstruction in \(\mathop{\mathrm{Ext}}\nolimits^2_\mathfrak{g}(V,V)=H^2(\mathfrak{g},\mathrm{End}_\mathbf{k}(V))\).

48.2 Levi decomposition↩︎

49 The third fundamental theorem of Lie theory↩︎

49.1 Exponentiating nilpotent and solvable Lie algebras and the third fundamental theorem of Lie theory↩︎

The following theorem implies the third fundamental theorem of Lie theory for solvable Lie algebras. Let \(\mathfrak{g}\) be a finite dimensional solvable Lie algebra over \(\Bbb K=\Bbb R\) or \(\Bbb C\) of dimension \(n\).

49.2 Formal groups↩︎

The third fundamental theorem of Lie theory assigns a simply connected Lie group \(G\) to any finite dimensional Lie algebra \(\mathfrak{g}\) over \(\Bbb R\) or \(\Bbb C\), such that \({\rm Lie}G=\mathfrak{g}\). But what about infinite dimensional Lie algebras? There are some examples when this is possible, for instance for \(\mathfrak{g}={\rm Vect}(M)\), the Lie algebra of vector fields for a smooth manifold \(M\), we can take \(G\) to be the universal cover of \({\rm Diff}_0(M)\), the group of diffeomorphisms of \(M\) homotopic to the identity, and for \(\mathfrak{g}=C^\infty(S^1,\mathfrak{k})\) for a finite dimensional Lie algebra \(\mathfrak{k}\) we can take \(G=C^\infty(S^1,K)\), where \(K\) is the simply connected Lie group corresponding to \(\mathfrak{k}\) (although we would need to explain in what sense \(G\) is a Lie group and \({\rm Lie}G=\mathfrak{g}\)). However, for a general infinite dimensional \(\mathfrak{g}\), such an assignment is typically impossible and a suitable group \(G\) does not exist.

However, this assignment becomes possible (and in fact not just over \(\Bbb R\) and \(\Bbb C\) but over any field of characteristic zero) if we replace the notion of a Lie group with a purely algebraic notion of a formal group. Roughly speaking, the notion of a formal group is the analog of the notion of a real or complex analytic Lie group where analytic functions are replaced by formal power series, and we don’t worry about their convergence. This allows us to work with infinite dimensional Lie algebras and over arbitrary fields of characteristic zero.

Let us give a precise definition. Given a vector space \(V\) over a field \({\boldsymbol{k}}\) of characteristic zero, define the algebra \({\boldsymbol{k}}[[V]]\) of formal regular functions on \(V\) to be \((SV)^*\), the dual of the symmetric algebra of \(V\). Since \(SV\) has a bialgebra structure \(\Delta_0: SV\to SV\otimes SV\) defined by \(\Delta_0(v)=v\otimes 1+1\otimes v\) for \(v\in V\), the dual map \(\Delta_0^*\) gives a commutative associative product on \({\boldsymbol{k}}[[V]]\), which is continuous in the weak topology of the dual space.³⁶ If \(x_i,i\in I\) is a linear coordinate system on \(V\) corresponding to a basis \(v_i,i\in I\), then we have a natural identification \({\boldsymbol{k}}[[V]]\cong {\boldsymbol{k}}[[x_i,i\in I]]\) of \({\boldsymbol{k}}[[V]]\) with the algebra of formal power series in \(x_i\). Note that here \(I\) can be a set of any cardinality, not necessarily finite or countable. Moreover, if \(\dim V<\infty\) then \({\boldsymbol{k}}[[V]]=\prod_{n\ge 0}S^nV^*\).

Finally, note that we have the augmentation homomorphism (counit) \(\varepsilon: {\boldsymbol{k}}[[V]]\to {\boldsymbol{k}}\) given by \(\varepsilon(f)=f(0)\), i.e., obtained by taking the quotient by the maximal ideal \(\mathfrak{m}\subset {\boldsymbol{k}}[[V]]\).

A (homo)morphism of formal groups \(\phi: G_1\to G_2\) is a (continuous) algebra homomorphism \(\mathcal{O}(G_2)\to \mathcal{O}(G_1)\) preserving the coproduct.³⁸

For example, a 1-dimensional formal group is defined by a power series \(F(x,y)\in {\boldsymbol{k}}[[x,y]]\), \(F(x,y)=x+y+...\), where ... denotes quadratic and higher terms, which is associative: \[F(F(x,y),z)=F(x,F(y,z)).\] Such a series \(F\) is called a formal group law. Namely, the map \(\Delta: {\boldsymbol{k}}[[x]]\to {\boldsymbol{k}}[[x_1,x_2]]\) is defined by the formula \[\Delta(f)(x_1,x_2)=f(F(x_1,x_2)).\] Higher-dimensional formal groups \(G\) can also be presented in this way, with \(F,x,y\) being vectors with \(\dim G\) entries rather than scalars.

So what does it have to do with groups? In fact, a lot: if \(G\) is a formal group then it defines a functor from the category of local commutative finite dimensional \({\boldsymbol{k}}\)-algebras to the category of groups, \[R\mapsto G(R)=\mathrm{Hom}_{\rm continuous}(\mathcal{O}(G),R)\] (where the topology on \(R\) is discrete).³⁹ Namely, the group law on such homomorphisms is defined by \[(a\circ b)(f)=(a\otimes b)(\Delta(f)).\] This makes sense even though \(\Delta(f)\) does not belong to \({\boldsymbol{k}}[[V]]\otimes {\boldsymbol{k}}[[V]]\) but only to its completion \({\boldsymbol{k}}[[V\oplus V]]\) since \(R\) is finite dimensional.

Moreover, any (homo)morphism of formal groups \(G_1\to G_2\) defines a morphism of functors \(G_1(?)\to G_2(?)\), and this assignment is compatible with composition. Furthermore, it is not hard to show that this assignment can be inverted, which allows us to define formal groups as representable functors from local finite dimensional commutative algebras to groups.

Any formal group \(G\) defines a Lie algebra \({\rm Lie}G\), which as a vector space is the continuous dual \(\mathfrak{g}:=({\mathfrak{m}}/{\mathfrak{m}^2})^*\). In other words, it is the underlying vector space \(V\) of \(G=(V,\Delta)\). Note that by compatibility of \(\Delta\) with \(\varepsilon\), for \(f\in \mathfrak m\) the element \(\Delta(f)-f\otimes 1-1\otimes f\) belongs to the completed tensor product \(\mathfrak{m}\widehat \otimes \mathfrak{m}\), thus projects to a well defined element of \((\mathfrak{g}\otimes \mathfrak{g})^*\). Thus the same is true for the element \(\Delta(f)-\Delta^{\rm op}(f)\) (where \(\Delta^{\rm op}\) is obtained from \(\Delta\) by swapping components); in fact, it defines an element of \((\wedge^2\mathfrak{g})^*\). Moreover, this element only depends on the residue \(\overline{f}\) of \(f\) in \(\mathfrak{g}^*= {\mathfrak{m}}/{\mathfrak{m}^2}\) (check it!). Denote the projection of \(\Delta(f)-\Delta^{\rm op}(f)\) to \((\wedge^2\mathfrak{g})^*\) by \(\delta(\overline{f})\). Then \(\delta: \mathfrak{g}^*\to (\wedge^2\mathfrak{g})^*\) is continuous, so it is dual to the map \([,]=\delta^*: \wedge^2\mathfrak{g}\to \mathfrak{g}\), and it is easy to show that \([,]\) is a Lie bracket on \(\mathfrak{g}\); namely, the Jacobi identity follows from the coassociativity of \(\Delta\) (check it!).

Conversely, given a Lie algebra \(\mathfrak{g}\) over \({\boldsymbol{k}}\) (not necessarily finite dimensional), we can use the Baker-Campbell-Hausdorff formula (Subsection 14.3) to assign a formal group to \(\mathfrak{g}\). Namely, take \(V=\mathfrak{g}\) and define \(\Delta: {\boldsymbol{k}}[[\mathfrak{g}]]\to {\boldsymbol{k}}[[\mathfrak{g}\oplus \mathfrak{g}]]\) by \[\Delta(f)(x,y)=f(\mu(x,y)),\] where \(\mu(x,y)=x+y+\frac{1}{2}[x,y]+...\) is the Baker-Campbell-Hausdorff series. Then the coassociativity of \(\Delta\) follows from the associativity of \(\mu\). In other words, we define \(G\) by setting its formal group law \(F\) to be equal to \(\mu\).

Over a field of positive characteristic (or over a commutative ring, such as \(\Bbb Z\)), much, but not all, of this story extends; let us for simplicity consider the finite dimensional case over a field. Namely, the definition of a formal group structure (say, on a finite dimensional space) is the same: it’s a coproduct on \({\boldsymbol{k}}[[x_1,...,x_n]]\) with the same properties as above.⁴⁰ The definition of the Lie algebra of a formal group also goes along for the ride. However, the reverse assignment fails, since the series \(\mu(x,y)\) is only defined over \(\Bbb Q\) and has all primes occurring in denominators of its coefficients. As a result, not any Lie algebra gives rise to a formal group, and the fundamental theorems of Lie theory for formal groups don’t hold.

In particular, there are many non-isomorphic 1-dimensional formal groups. For example, we have the additive group law \(F(x,y)=x+y\) as above, but also the multiplicative group law \(F(x,y)=x+y+xy\), which is called so because this means that \(1+F(x,y)=(1+x)(1+y)\). In characteristic zero these are isomorphic by the map \[x\mapsto e^x-1=\sum_{n\ge 1}\frac{x^n}{n!},\] (not surprisingly in view of Corollary 79), but in positive characteristic this series does not make sense and in fact the additive and multiplicative formal groups are not isomorphic (check it!). There are also many other 1-dimensional formal group laws, commutative and not. Such (commutative) formal group laws are very important in algebraic topology, since they parametrize (complex-oriented) cohomology theories. For example, the additive group law corresponds to ordinary cohomology and the multiplicative one to \(K\)-theory. In characteristic zero the isomorphism between the additive and multiplicative formal groups leads to the Chern character map which identifies cohomology and \(K\)-theory of a topological space with \(\Bbb Q\)-coefficients.

50 Ado’s theorem↩︎

50.1 The nilradical↩︎

Consider now a solvable Lie algebra \(\mathfrak{a}\) over \(\Bbb C\) and its adjoint representation. By Lie’s theorem, in some basis \(\mathfrak{a}\) acts in this representation by upper triangular matrices. Let \(\mathfrak{n}\subset \mathfrak{a}\) be the subset of nilpotent elements (the nilradical of \(\mathfrak{a}\)). Thus \(\mathfrak{n}\) is the set of \(x\in \mathfrak{a}\) that act in this basis by strictly upper triangular matrices. In particular, \(\mathfrak{n}\supset [\mathfrak{a},\mathfrak{a}]\), so \(\mathfrak{a}/\mathfrak{n}\) is abelian.

50.2 Algebraic Lie algebras↩︎

Let us say that a finite dimensional complex Lie algebra \(\mathfrak{g}\) is algebraic if \(\mathfrak{g}\) is the Lie algebra of a group \(G=K\ltimes N\), where \(K\) is a reductive group and \(N\) a unipotent group. It turns out that this is equivalent to being the Lie algebra of an affine algebraic group over \(\Bbb C\) (i.e., a closed subgroup in \(GL_n(\Bbb C)\) defined by polynomial equations), which motivates the terminology.

A finite dimensional complex Lie algebra need not be algebraic:

Nevertheless, we have the following proposition.

50.3 Faithful representations of nilpotent Lie algebras↩︎

Let \(\mathfrak{n}\) be a finite dimensional nilpotent Lie algebra over \(\Bbb C\). In this subsection we will show that \(\mathfrak{n}\) has a finite dimensional faithful representation.

To this end, recall that by Theorem 321, \(\mathfrak{n}={\rm Lie}(N)\) where \(N\) is a simply connected Lie group, and the exponential map \(\exp: \mathfrak{n}\to N\) is bijective. Moreover, the multiplication law of \(N\), when rewritten on \(\mathfrak{n}\) using the exponential map, is given by polynomials.

50.4 Faithful representations of general finite dimensional Lie algebras↩︎

51 Borel subgroups and the flag manifold of a complex reductive Lie group↩︎

51.1 Borel subgroups and subalgebras↩︎

Let \(G\) be a connected complex reductive Lie group, \(\mathfrak{g}={\rm Lie}(G)\). Fix a Cartan subalgebra \(\mathfrak{h}\subset \mathfrak{g}\) with a system of simple positive roots \(\Pi\), and consider the corresponding triangular decomposition \(\mathfrak{g}=\mathfrak{n}_-\oplus\mathfrak{h}\oplus \mathfrak{n}_+\), where \(\mathfrak{n}_+\) is spanned by positive root elements and \(\mathfrak{n}_-\) by negative root elements. Let \(H\) be the maximal torus in \(G\) corresponding to \(\mathfrak{h}\), \(N_+\) the unipotent subgroup of \(G\) corresponding to \(\mathfrak{n}_+\), and \(B_+=HN_+\) the solvable subgroup with \({\rm Lie}(B_+)=\mathfrak{b}_+:=\mathfrak{h}\oplus \mathfrak{n}_+\); these are all closed Lie subgroups.

Since all pairs \((\mathfrak{h},\Pi)\) are conjugate, this definition does not depend on the choice of \((\mathfrak{h},\Pi)\).

51.2 The flag manifold of a connected complex reductive group↩︎

Thus the set of all Borel subalgebras (or subgroups) in \(G\) is the homogeneous space \(G/B_+\), a complex manifold. It is called the flag manifold of \(G\). Note that it only depends on the semisimple part \(\mathfrak{g}_{ss}\subset \mathfrak{g}\) and does not depend on the choice of the Cartan subalgebra and triangular decomposition.

Let \(G^c\subset G\) be the compact form of \(G\), with Lie algebra \(\mathfrak{g}^c\subset \mathfrak{g}\). It is easy to see that \(\mathfrak{g}^c+\mathfrak{b}_+=\mathfrak{g}\). Thus the \(G^c\)-orbit \(G^c\cdot 1\) of \(1\in G/B_+\) contains a neighborhood of \(1\) in \(G/B_+\). Hence the same holds for any point of this orbit, i.e., \(G^c\cdot 1\subset G/B_+\) is an open subset. But it is also compact, since \(G^c\) is compact, hence closed. As \(G/B_+\) is connected, we get that \(G^c\cdot 1=G/B_+\), i.e., \(G^c\) acts transitively on \(G/B_+\).

Also the Cartan involution \(\omega\) maps positive root elements to negative ones, so \(G^c\cap B_+\subset w_0(B_+)\cap B_+=H\). Thus \(G^c\cap B_+=H^c\), a maximal torus in \(G^c\). So we get

Another realization of the flag manifold is as the \(G\)-orbit of the line spanned by the highest weight vector in an irreducible representation with a regular highest weight. Namely, let \(\lambda\in P_+\) be a dominant integral weight with \(\lambda(h_i)\ge 1\) for all \(i\) (i.e., \(\lambda=\mu+\rho\) for \(\mu\in P_+\)). Let \(L_\lambda\) be the corresponding irreducible representation with highest weight vector \(v_\lambda\). We have \(\mathfrak{b}_+\cdot \Bbb Cv_\lambda=\Bbb Cv_\lambda\), but \(e_{-\alpha} v_\lambda\ne 0\) for any \(\alpha\in R_+\) (as \(e_\alpha e_{-\alpha}v_\lambda=h_\alpha v_\lambda=(\lambda,\alpha^\vee)v_\lambda\), and \((\lambda,\alpha^\vee)>0\)). Moreover, these vectors have different weights, so are linearly independent. Thus \(\mathfrak{b}_+\) is the stabilizer of \(\Bbb Cv_\lambda\) in \(\mathfrak{g}\). Hence any \(g\in G\) which preserves \(\Bbb Cv_\lambda\) belongs to the normalizer of \(\mathfrak{b}_+\) (or, equivalently, \(B_+\)), i.e., \(g\in B_+\). Thus \(\mathcal{O}:=G\cdot \Bbb Cv_\lambda\subset \Bbb P L_\lambda\) is identified with \(G/B_+\). This shows that \(\mathcal{O}\) is compact, hence closed, i.e., \(\mathcal{O}=G/B_+\) is a smooth complex projective variety.

Let \(A=\exp(i\mathfrak{h}^c)\subset H\), \(K=G^c\), \(N=N_+\). Proposition 329 immediately implies

A similar theorem holds for real reductive groups (Theorem 334).

51.3 The Borel fixed point theorem↩︎

Let \(V\) be a finite dimensional representation of a finite dimensional \(\Bbb C\)-Lie algebra \(\mathfrak{a}\), and \(X\subset \Bbb PV\) be a subset. We will say that \(X\) is \(\mathfrak{a}\)-invariant (or fixed by \(\mathfrak{a}\)) if it is \(\exp(\mathfrak{a})\)-invariant.

Note that for \(X=\Bbb PV\), the Borel fixed point theorem reduces to Lie’s theorem (Theorem 99).

51.4 Parabolic and Levi subalgebras↩︎

A Lie subalgebra \(\mathfrak{p}\supset \mathfrak{b}\) of a reductive Lie algebra \(\mathfrak{g}\) containing some Borel subalgebra \(\mathfrak{b}\subset \mathfrak{g}\) is called a parabolic subalgebra of \(\mathfrak{g}\). The corresponding connected Lie subgroup \(P\subset G\) is called a parabolic subgroup. It is easy to see that \(P\subset G\) is necessarily closed (check it!).

Let \(P\subset G\) be a parabolic subgroup with Lie algebra \(\mathfrak{p}\). Let \(\mathfrak{u}\subset \mathfrak{p}\) be the nilpotent radical of \(\mathfrak{p}\); for instance, if \(\mathfrak{p}\supset \mathfrak{b}_+\) then \(\mathfrak{u}\) is the Lie subalgebra spanned by \(e_\alpha\) such that \(e_{-\alpha}\notin \mathfrak{p}\). It is easy to see that there exists a (non-unique) Lie subalgebra \(\mathfrak l\subset \mathfrak{p}\) complementary to \(\mathfrak{u}\), which therefore projects isomorphically to \(\mathfrak{p}/\mathfrak{u}\); indeed, if \(\mathfrak{p}\supset \mathfrak{b}_+\) then we can take \(\mathfrak l\) to be the Lie subalgebra spanned by \(\mathfrak{h}\) and \(e_\alpha,e_{-\alpha}\) where \(\alpha\) runs through positive roots for which \(e_{-\alpha}\in \mathfrak{p}\). Such a subalgebra \(\mathfrak l\) is called a Levi subalgebra of \(\mathfrak{p}\), and we have \(\mathfrak{p}=\mathfrak l\ltimes \mathfrak{u}\), which is \(\mathfrak l\oplus \mathfrak{u}\) as a vector space.

Let \(U=\exp(\mathfrak{u})\). The quotient \(P/U\) is a reductive group with Lie algebra \(\mathfrak{p}/\mathfrak{u}\). A Levi subgroup of \(P\) is a subgroup \(L\) in \(P\) such that \(\mathfrak l:={\rm Lie}(L)\) is a Levi subalgebra of \(\mathfrak{p}\); equivalently, \(L\) projects isomorphically to \(P/U\), so we have \(P=L\ltimes U\), written shortly as \(P=LU\). It is not difficult to show that all Levi subgroups of \(P\) (or, equivalently, all Levi subalgebras of \(\mathfrak{p}\)) are conjugate by the action of \(U\) (check it!).

For example, \(L\) is a maximal torus if and only if \(P\) is a Borel subgroup, and \(L=G\) if and only if \(P=G\).

51.5 Maximal solvable and maximal nilpotent subalgebras↩︎

Note that \(\mathfrak{b}_+\) is a maximal solvable subalgebra of \(\mathfrak{g}\); indeed, any bigger parabolic subalgebra contains a negative root vector, hence the corresponding root \(\mathfrak{sl}_2\)-subalgebra, so it is not solvable. Moreover, \(B_+\) is a maximal solvable subgroup of \(G\): if \(P\supset B_+\) then some element \(g\in P\) does not normalize \(\mathfrak{b}_+\), so \({\rm Lie}(P)\) has to be larger than \(\mathfrak{b}_+\), hence not solvable. Thus any Borel subalgebra (subgroup) is a maximal solvable one. It turns out that the converse also holds.

Let us say that a Lie subalgebra \(\mathfrak{a}\subset \mathfrak{g}\) is a nilpotent subalgebra if it consists of nilpotent elements. Note that this is a stronger condition than just being nilpotent as a Lie algebra; for example, a Cartan subalgebra is a nilpotent Lie algebra (since it is abelian) but it is not a nilpotent subalgebra of \(\mathfrak{g}\).

A similar result holds for groups, with the same proof:

We also have

51.6 Iwasawa decomposition of a real semisimple linear group↩︎

Let \(G_\theta=K^cP_\theta\) be the polar decomposition of a real form of a complex semisimple group \(G\), \(\mathfrak{g}_\theta=\mathfrak{k}^c\oplus \mathfrak{p}_\theta\) the additive version, \(\mathfrak{a}\subset \mathfrak{p}_\theta\) a maximal abelian subspace. Let \(A=\exp(\mathfrak{a})\subset P_\theta\) be the corresponding abelian subgroup of \(G_\theta\). Pick a generic element \(a\in \mathfrak{a}\). Let \(\mathfrak{z}=\mathfrak{g}_\theta^a\) be the centralizer of \(a\) in \(\mathfrak{g}_\theta\) and let \(\mathfrak{n}_{a,\pm}\) be the (nilpotent) Lie subalgebras of \(\mathfrak{g}_\theta\) spanned by eigenvectors of \({\rm ad}a\) with positive, respectively negative eigenvalues, so that \(\mathfrak{g}_\theta=\mathfrak{n}_{a-}\oplus \mathfrak{z}\oplus \mathfrak{n}_{a+}\). Let \(N_{a\pm}=\exp(\mathfrak{n}_{a\pm})\).

The following theorem is a generalization of Proposition 81.

Theorem 334 is proved in the following exercise.

51.7 The Bruhat decomposition↩︎

Let \(G\) be a connected complex reductive group, \(H\subset G\) a maximal torus, \(B=B_+\supset H\) a Borel subgroup. The Bruhat decomposition is the decomposition of \(G\) into double cosets of \(B\).

Let \(N(H)\) be the normalizer of \(H\) in \(G\) and \(W=N(H)/H\) be the Weyl group. Given \(w\in W\), let \(\widetilde{w}\) be a lift of \(w\) to \(N(H)\) and consider the double coset \(B\widetilde{w} B\subset G\). Since any two lifts of \(w\) differ by an element of \(H\) which is contained in \(B\), the set \(B\widetilde{w} B\) does not depend on the choice of \(\widetilde{w}\), so we will denote it by \(BwB\).

Theorem 337 can be reformulated as a classification of \(B\)-orbits on the flag manifold \(G/B\). Namely, given \(w\in W\), the set \(BwB/B\) is an orbit of \(B\) on \(G/B\), which we will denote by \(C_w\). By Theorem 336, \(C_w\) are disjoint, and Theorem 337 is equivalent to

The sets \(BwB\) are called Bruhat cells and the sets \(C_w\) are called Schubert cells.⁴²

Note that for type \(A_{n-1}\) (\(G=SL_n(\Bbb C)\) or its quotient), we have already proved Theorem 338 in Subsection 47.3, where we decomposed the flag manifold \(\mathcal{F}_n\) into Schubert cells labeled by permutations.

A proof of Theorem 338 can be found, for example, in the textbook [14]. It is also sketched in the following exercise.

Let us derive some corollaries of Theorem 338.

References↩︎

---

1. During the first semester and at the beginning of the second one homework problems were also assigned from [1].↩︎

2. More precisely, for \(C^k\) and real analytic manifolds regular functions will be assumed real-valued, unless specified otherwise. In the complex analytic case there is, of course, no choice, and regular functions are automatically complex-valued.↩︎

3. Recall that a subset \(Z\) of a topological space \(X\) is called locally closed if it is a closed subset in an open subset \(U\subset X\). It is clear that embedded submanifolds are locally closed. For this reason they are often called locally closed (embedded) submanifolds.↩︎

4. In other words, using a fancier language, \(\boldsymbol{v}\) defines a derivation of the sheaf of regular functions on \(X\).↩︎

5. For brevity for \(g\in G\), \(x\in \mathfrak{g}\) we denote \(L_gx\) by \(gx\) and \(R_gx\) by \(xg\).↩︎

6. Note that this Lie algebra is infinite dimensional for all real manifolds and many (but not all) complex manifolds of positive dimension.↩︎

7. Although we claimed in Theorem 13 that a closed subgroup of a Lie group is always a Lie subgroup, we did not prove it, so we need to prove it in this case.↩︎

8. Note that this definition is not quite equivalent to Definition 3.51 in [1] of the same notion, which is less conventional. For example, according to the definition of [1], every complex elliptic curve has a real form, which does not agree with the definition from algebraic geometry (cf. Example 23).↩︎

9. An exception is the adjoint representation of a real Lie group and associated tensor representations, which are real.↩︎

10. The grading on \(T\mathfrak{g}\) does not descend to \(U(\mathfrak{g})\), in general, since the relation \(xy-yx=[x,y]\) is not homogeneous: the right hand side has degree 1 while the left hand side has degree 2. So \(U(\mathfrak{g})\) is not graded but is only filtered.↩︎

11. In fact, we will see later that over an algebraically closed field of characteristic zero, a finite dimensional Lie algebra consisting of semisimple elements is automatically abelian.↩︎

12. Usually one also adds the condition that \(V\) is a finitely generated \(U(\mathfrak{g})\)-module, but we don’t need this condition here, so we won’t impose it.↩︎

13. This also gives another proof of the fact that \(A\) is a direct sum of matrix algebras, i.e. complete reducibility of \(V^{\otimes N}\).↩︎

14. The maximal coroot \(\theta^\vee\) should not be confused with the coroot \(\widetilde{\theta}^\vee\) corresponding to the maximal root \(\theta\) (highest weight of the adjoint representation) under a \(W\)-invariant identification \(\mathfrak{h}^*\cong \mathfrak{h}\). In the non-simply-laced case they are not even proportional: e.g., for the root system \(B_2\), \(\theta^\vee=(1,1)\) while \(\widetilde{\theta}^\vee=(2,0)\). This may be confusing since according to the general coroot notation, \(\widetilde{\theta}^\vee\) should be denoted by \(\theta^\vee\). ↩︎

15. The converse statement also holds, but we will not need it.↩︎

16. Cantor’s theorem says that any continuous function on a compact set \(X\) is uniformly continuous.↩︎

17. Note that a finite Borel measure on a compact Hausdorff space with a countable base is necessarily regular.↩︎

18. Recall that \(\psi\) determines the probability \(p(U,t)\) to find the electron in a region \(U\subset \Bbb R^3\) at a time \(t\), which is given by the formula \(p(U,t)=\int_U |\psi(x,y,z,t)|^2dxdydz\).↩︎

19. In the classical mechanics setting, the existence of this conservation law is the reason why orbits for Coulomb potential are periodic (Kepler’s law), while this is not so for other rotationally invariant potentials, except harmonic oscillator. It was discovered many times over the last 300 years. This is one of the most basic examples of “hidden symmetry".↩︎

20. The element \(\widetilde{w}\) in general depends on the decomposition of \(w\) as a product of simple reflections. One can show it does not if we take only reduced decompositions, but we will not need this.↩︎

21. For classical Lie algebras the set \(\boldsymbol{K}\) will be computed explicitly in Exercise 290. The explicit answer is known for exceptional Lie algebras as well, but we will not discuss it here.↩︎

22. The advantage of passing from \(g\) to \(\theta\) is that the equation \(\theta^2=1\) is much easier to solve than \(g\omega(g)=1\), as it just means that we have a decomposition of \(\mathfrak{g}\) into the \(+1\)- and \(-1\)-eigenspaces of \(\theta\).↩︎

23. We say that a connected Lie group \(G\) is simple if so is its Lie algebra. Thus this does not quite mean that \(G\) is simple as an abstract group: it may have a finite center (e.g., \(G=SU(2)\) or \(SL_2(\Bbb C)\)). For this reason such “simple" groups are sometimes called almost simple. However, the corresponding adjoint group \(G_{\rm ad}\) is indeed simple as an abstract group.↩︎

24. Of course, the group \(K^c\) depends on \(\theta\), but for simplicity we will not indicate this dependence in the notation.↩︎

25. Here is another proof of this corollary. Let \(B(x,y)\) be the Killing form of \(\mathfrak{g}^c\). Since \(B\) is negative definite, the form \(-B\) extends to a bi-invariant Riemannian metric on \(G_c\). Since \(G^c\) is compact, the Hopf-Rinow theorem guarantees that for any \(g\in G^c\) there is a geodesic on \(G^c\) in this metric connecting \(1\) and \(g\). But it is easy to see that this geodesic is a segment of a one-parameter subgroup of \(G^c\), which implies the statement.↩︎

26. This factorization is not unique.↩︎

27. Note that the function \(\rho(t)\) may be multivalued, but its branches differ from each other by a root of unity, so the function \(|\Delta(t)|\) is well defined. Namely, \(|\Delta(t)|=|\Delta_0(t)|\) where \(\Delta_0(t)=\prod_{\alpha\in R^+}(\alpha(t)-1)\).↩︎

28. Note that if \(\mathfrak{g}\) is finite dimensional then \(\wedge^i\mathfrak{g}^*=(\wedge^i\mathfrak{g})^*\).↩︎

29. Note that \(H^1(\mathfrak{g})\) already appeared earlier in Section 18.↩︎

30. Moreover, we have an algebra homomorphism \(S: R\to R\) induced by the inversion map \(G\to G\) called the antipode. This makes \(R\) into what is called a graded Hopf algebra.↩︎

31. A similar idea can be used to find the cohomology of \({\rm Spin}(n)\) (see Exercise 306 below) but it is a bit more complicated since there is no cell decomposition with zero boundary map, and thus any cell decomposition has strictly more than \(2^r\) cells for sufficiently large \(n\) (as there is 2-torsion in the integral cohomology).↩︎

32. Note that setting \(q=1\) in the \(q\)-binomial theorem, we get the familiar formula from calculus, often called the binomial theorem: \[\sum_{n\ge 0}\binom{m+n}{m}z^n= \frac{1}{(1-z)^{m+1}}.\]↩︎

33. Note that \(H^1(\mathfrak{g},V)\) appeared earlier in Section 18 and Whitehead’s theorem in the case of \(H^1\) was proved in Subsection 18.2.↩︎

34. In other words, we have reduced to the case when \({\rm rad}(\mathfrak{g})=V\) is abelian, and we have shown above that abelian extensions are classified by \(H^2(\mathfrak{g}_{ss},V)\), which is zero by Whitehead’s theorem.↩︎

35. The reason for this terminology is that these groups act by unipotent operators on the adjoint representation.↩︎

36. Recall that if \(E\) is a vector space then the dual space \(E^*\) carries the weak topology whose basis of neighborhoods of zero is given by orthogonal complements of finite dimensional subspaces of \(E\).↩︎

37. Note that if \(\dim V<\infty\), any such homomorphism is automatically continuous.↩︎

38. Thus we forget the linear structure on \(V\) (it does not have to be preserved by homomorphisms). In other words, to specify a formal group, we don’t need to specify a vector space \(V\) but only need to specify a (topological) ring isomorphic to \({\boldsymbol{k}}[[V]]\) for some \(V\) and equipped with a coproduct.↩︎

39. Again, continuity is automatic if \(\dim G<\infty\).↩︎

40. More precisely, instead of \(SV\) we should take the symmetric algebra with divided powers \(\Gamma V\), defined by \(\Gamma^mV:=(S^mV^*)^*\). Note that in characteristic \(p\), \(\Gamma^mV\) is not naturally isomorphic to \(S^mV\) for \(m\ge p\).↩︎

41. Here is another proof of this proposition. The one-parameter group \(e^{td}\) of automorphisms of \(\mathfrak{a}\) preserves the set of characters of \(\mathfrak{a}\) occurring in its adjoint representation. Hence it must preserve each of them individually, as there are finitely many and this group is connected. But by definition of \(\mathfrak{n}\) these characters span \((\mathfrak{a}/\mathfrak{n})^*\). Thus \(d\) acts trivially on \(\mathfrak{a}/\mathfrak{n}\).↩︎

42. We note that Bruhat cells, unlike Schubert cells, are not literally cells in the topological sense – they are not homeomorphic to an affine space, but are homeomorphic to the product of an affine space and a torus.↩︎