June 04, 2026
The goal of this article is to show the categorical links between on the one hand the category of reductive Lie algebra pairs \(\mathcal{RLP}\) and on the other hand the category of Lie-Yamaguti algebras \(\mathcal{LY}\). The fact that the well-known construction of an enveloping algebra associating to a Lie-Yamaguti algebra a reductive Lie algebra pair is not functorial leads us to the main construction of the article, namely a left adjoint to the natural restriction functor \(G:\mathcal{RLP}\to\mathcal{LY}\). As a final result we observe that the construction of the enveloping algebra becomes functorial when one restricts the morphisms of the categories \(\mathcal{RLP}\) and \(\mathcal{LY}\) to the surjective ones. Then it becomes a right adjoint to the restriction functor.
Lie algebras, Lie-Yamaguti algebra , reductive Lie algebra pair, homogeneous spaces, category theory, Lie triple systems, Leibniz algebras.
17A30, 17A32, 17A40, 18A40, 17B05, 22F30, 53B05.
Dedicated to the memory of Otto H. Kegel.
Otto H. Kegel has written most of his papers in group theory, but he was also quite open to other algebraic subjects like for instance associative algebras and Lie algebras: during the Master thesis of one of the authors, M.B., about certain Lie
algebras, he used to say that established properties of groups can very often be copied to other fields.
In this note, we shall be dealing with a categorical relation between the class of reductive Lie algebra pairs \(\mathcal{RLP}\) and the class of Lie-Yamaguti algebras \(\mathcal{LY}\). The first class comes from a Lie theoretical study of reductive homogeneous spaces (mostly in the 1950’s, see e.g. [1], [2]), and consists of triples \(\big(\mathfrak{g},\mathfrak{h},\mathfrak{m}\big)\) consisting
of a Lie algebra \(\mathfrak{g}\), a subalgebra \(\mathfrak{h}\subset\mathfrak{g}\), and an \(\mathrm{ad}_\mathfrak{h}\) invariant subspace \(\mathfrak{m}\subset\mathfrak{g}\) such that \(\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}\): note that in that definition ‘reductive’ does not mean that one of the two Lie
algebras \(\mathfrak{g}\) or \(\mathfrak{h}\) is a reductive Lie algbra in the sense that its adjoint representation is completely reducible. The second class is K. Yamaguti’s generalization
(see [3]) of N. Jacobson’s Lie triple systems [4] to a vector
space (or even module) \(E\) (interpreted as the tangent space at the distinguished point of the homogeneous space) equipped with a bilinear operation \(T\) (representing the differential
geometric torsion) and a trilinear operation \(R\) (representing the differential geometric curvature) of the canonical connection on a reductive homogeneous space, see e.g. [2]. Both operations \(T\) and \(R\) satisfy six identities derived from the classical Bianchi identities, see e.g. [5]. This latter approach can be seen as an redundancy-free description of the affine geometry of the homogeneous space, but is computationally more involved. Both classes \(\mathcal{RLP}\) and \(\mathcal{LY}\) are categories whose morphisms are Lie algebra morphisms preserving the splitting and linear maps intertwining the bilinear and trilinear operations,
respectively. Lie-Yamaguti algebras have been studied in numerous works more recently, see [6] (from the operadic point of view) and [7], and [8] for simple algebras in relation to Jordan algebras. The aim of this note is to
study and establish a functorial relation between the two categories \(\mathcal{RLP}\) and \(\mathcal{LY}\): such a link –controlled by category theory– can be quite advantageous in
order to translate or copy properties or concepts between the two, and the recent operadic approaches to classes of nonassociative algebras, see e.g. [6], are
formulated in this language.
First, there is a well-known ‘easy’ functor \(\mathbf{G}':\mathcal{RLP}\to \mathcal{LY}\) sending an object \(\big(\mathfrak{g},\mathfrak{h},\mathfrak{m}\big)\) in \(\mathcal{RLP}\) to the complement \(\mathfrak{m}\) and projecting (iterated) Lie brackets to the subspace and the subalgebra to obtain the bilinear and trilinear operations of a Lie Yamaguti
algebra.
Secondly, it turns out to be much more difficult to find a functor \(\mathbf{F'}\) in the other direction \(\mathcal{LY}\to \mathcal{RLP}\): the problem we faced is the fact that to each
Lie-Yamaguti algebra \(E\) one can assign a well-known classical reductive Lie algebra pair, its so-called enveloping Lie algebra, see e.g. [3] or [9], \(\hat{\mathfrak{g}}(E)\) –which is a reductive Lie algebra pair with \(\mathfrak{m}=E\) plus the holonomy of the connection, see e.g. [5] for definitions. However, as it was pointed out to the authors by
Yannick Voglaire, this assignment is well-known to be NOT functorial for the above categories, and we shall provide a simple explicit example of this defect in Section 2 of this note.
The first main result of this work is the explicit construction of a left adjoint functor \(\mathbf{F}':\mathcal{LY}\to \mathcal{RLP}\) to the functor \(\mathbf{G}'\) in the general Lie Yamaguti case, see Theorem 1: this will define a universal construction, a sort of free reductive Lie
algebra pair \(\mathfrak{g}(E)\) generated by the Lie-Yamaguti algebra \(E\). However, this adjunction does not define a categorical equivalence. Our result generalizes such a
functorial construction for the particular case of Lie triple systems which had already been sketched without details in [4] –around ten years before D. Kan’s
definition of adjoint functors– and finally duly formulated in [10].
Next, in order to reduce a little bit the above redundancy in the category \(\mathcal{RLP}\) we have found it useful to define the subcategory \(\mathrm{m}\mathcal{RLP}\) of all \(\mathfrak{m}\)-generated reductive Lie algebra pairs, i.e. where the Lie algebra \(\mathfrak{g}\) is generated by the subspace \(\mathfrak{m}\). It turns
out that the inclusion functor \(\mathbf{J}:\mathrm{m}\mathcal{RLP}\to\mathcal{RLP}\) has a right adjoint \(\mathfrak{i}:\mathcal{RLP}\to \mathrm{m}\mathcal{RLP}\) assigning to each \(\big(\mathfrak{g},\mathfrak{h},\mathfrak{m}\big)\) the well-known ideal \(\mathfrak{i}(\mathfrak{g}):=\mathfrak{m}+[\mathfrak{m},\mathfrak{m}]\subset \mathfrak{g}\), see e.g. [2] . Hence the subcategory is coreflective, see e.g. [11] for
definitions, like the subcategory of all torsion abelian groups in the category of all abelian groups. It is not hard to see that the above functors \(\mathbf{F'}\) and \(\mathbf{G'}\) factor over the subcategory, and we arrive at the following chain of adjunctions \[\begin{tikzcd} \mathcal{LY} \arrow[r, yshift=0.7ex, "\mathbf{F}"] &
m\mathcal{RLP}\arrow[l, yshift=-0.7ex, "\mathbf{G}"] \arrow[r, yshift=0.7ex, "\mathbf{J}"] &\mathcal{RLP}\arrow[l, yshift=-0.7ex, "\boldsymbol{\mathfrak{i}}"] \end{tikzcd}\] where \(\mathbf{F}'=\mathbf{JF}\) and \(\mathbf{G}'=\mathbf{G}\boldsymbol{\mathfrak{i}}\). The reductive Lie algebra pair \(\mathbf{F}E=\mathfrak{g}(E)\)
associated to a Lie-Yamaguti algebra \(E\) by the functor \(\mathbf{F}\) is an explicit quotient space of \(E\oplus \Lambda^2E\). It is curious and practical
at the same time that the intermediate computations use the theory of Leibniz algebras, see e.g. [12] for right Leibniz algebras, while here we use left Leibniz
algebras (see e.g. [13]), which have also been used in [9] for the
enveloping Lie algebra. The difference between \(\hat{\mathfrak{g}}(E)\) and \(\mathfrak{g}(E)\) can be seen in the corollary that an abelian Lie-Yamaguti algebra \(E\) has an abelian enveloping Lie algebra \(\hat{\mathfrak{g}}(E)=E\) whereas the functor \(\mathbf{F}\) sends it to the non abelian, Heisenberg type Lie algebra
\(\mathfrak{g}(E)=E\oplus \Lambda^2E\) with the obvious nonabelian Lie bracket.
The second main result is a possibility of situating the enveloping Lie algebra \(\hat{\mathfrak{g}}(E)\) of a Lie-Yamaguti algebra based on \(E\) in a categorical manner: we show in Theorem
2 that if we define subcategories \(\mathcal{LY}\mathrm{s}\), \(\mathrm{m}\mathcal{RLP}\mathrm{s}\), and \(\mathrm{m}\mathcal{RLP}\mathrm{s}\) of the above three categories by only allowing morphisms which are surjective linear maps, then
the assignment of an enveloping Lie algebra to a Lie-Yamaguti algebra extends to surjective morphisms and thus defines a covariant functor \(\hat{\mathfrak{g}}\) which turns out to be a right adjoint
functor of the restriction \(\mathbf{G}_\mathrm{s}\) of the functor \(\mathbf{G}\) to the subcategory \(\mathrm{m}\mathcal{RLP}\mathrm{s}\) of \(\mathrm{m}\mathcal{RLP}\). Note that these subcategories are respected by the two above functors \(\mathbf{F}\) and \(\mathbf{G}\). Hence there is the following
second adjunction which implies that every \(\mathfrak{m}\)-generated reductive Lie algebra pair \((\mathfrak{g},\mathfrak{h,\mathfrak{m}})\) is contained in a chain of central extensions:
\[\begin{tikzcd} \mathrm{m}\mathcal{RLP}\mathrm{s} \arrow[r, yshift=0.7ex, "\mathbf{G}_\mathrm{s}"] & \mathcal{LY}\mathrm{s} \arrow[l, yshift=-0.7ex, "\hat{\mathfrak{g}}"]
\end{tikzcd}
~~~\mathrm{implying}~~~
\begin{tikzcd} \mathfrak{g}(\mathfrak{m}) \arrow[r, twoheadrightarrow, "\boldsymbol{\epsilon}_\mathfrak{g}"] & \mathfrak{g} \arrow[r, twoheadrightarrow, "\boldsymbol{\eta}_{s\mathfrak{g}}"] &\hat{\mathfrak{g}}(\mathfrak{m})
\end{tikzcd}.\]
Acknowledgements: The authors would like to dedicate this work to the memory of Otto H. Kegel who has passed away last year on his \(91\)st birthday. We all remember him as a gentle, competent and very
generous mathematician whose support has always been unlimited and valuable. In particular I, M.B., have learned true mathematics in innumerous conversations with him when I did my Diplomarbeit (Master thesis): I profited a lot from his witty, but at the
same time precise way of looking at mathematics which shaped my way of mathematical thinking and allowed me to pass from theoretical physics to mathematics. He also helped me in a substantial way to find a job at a university through his wide-spread
network of colleagues and friends. It is a great loss that he is non longer with us.
The authors would also like to thank Yannick Voglaire for fruitful discussions, and the anonymous referee for giving us very many very useful remarks which improved the manuscript considerably.
In this section \(K\) is always a fixed commutative associative unital ring containing the field of all rational numbers \(\mathbb{Q}\) as a unital subring. All modules are considered over \(K\), and the symbol \(\otimes\) is short for \(\otimes_K\). In view of MacLane’s coherence theorem (see [11] we can and will assume that the cartesian product \(\times\) for sets and the tensor product \(\otimes\) are associative. We shall write \(K\)-multilinear maps in the old-fashioned non-tensorial way with multiple arguments separated by commas.
Recall that a reductive Lie algebra pair \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\), which in the following will be abbreviated by RLA pair, is a triple consisting of a Lie algebra \(\mathfrak{g}\), a subalgebra \(\mathfrak{h}\subset \mathfrak{g}\), and a complementary \(K\)-submodule \(\mathfrak{m}\subset
\mathfrak{g}\) of \(\mathfrak{h}\) in \(\mathfrak{g}\) such that \(\mathfrak{m}\) is invariant under the adjoint action of \(\mathfrak{h}\), i.e., \[\label{EqDefMStableAdH} [\mathfrak{h},\mathfrak{m}]\subset \mathfrak{m} ~~\Longleftrightarrow~~
\forall~y\in\mathfrak{h},~\forall~z\in \mathfrak{m}:~ [y,z]\in \mathfrak{m},\tag{1}\] see e.g. [14] for cohomological obstructions to the existence
of such complements. These pairs form the objects of a category \(\mathcal{RLP}\), where the set of all morphisms from \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) to \((\mathfrak{g}',\mathfrak{h}',\mathfrak{m}')\) consists of all morphisms of Lie algebras \(\phi:\mathfrak{g}\to
\mathfrak{g}'\) respecting the splitting, i.e., \(\phi(\mathfrak{h})\subset \mathfrak{h}'\) and \(\phi(\mathfrak{m})\subset \mathfrak{m}'\). The restrictions of \(\phi\) to \(\mathfrak{h}\) and to \(\mathfrak{m}\) are denoted by \(\phi_\mathfrak{h}\) and \(\phi_\mathfrak{m}\), respectively. Recall that the particular case \([\mathfrak{m},\mathfrak{m}]\subset\mathfrak{h}\) leads to \(\mathbb{Z}_2\)-graded Lie
algebras, also called symmetric Lie algebras, see e.g. [2] or [15].
Returning to the general case, for a given RLA pair \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) denote the canonical projections \(\mathfrak{g}\to \mathfrak{h}\) (with kernel \(\mathfrak{m}\)) and \(\mathfrak{g}\to \mathfrak{m}\) (with kernel \(\mathfrak{h}\)) by \[\forall~x\in \mathfrak{g}:~~~x\mapsto
x_\mathfrak{h}\in\mathfrak{h},~~~ \mathrm{and}~~~ x\mapsto x_\mathfrak{m}\in\mathfrak{m}\] where we adopt the notation from [2]. Obviously, we have \(x=x_\mathfrak{h}+x_\mathfrak{m}\). Moreover, as an immediate consequence of (1 ), we have the following identities: \[\label{EqDefRLAadHpreservesHAndM} \forall~x\in \mathfrak{g},~\forall~y\in \mathfrak{h},~\forall~z\in \mathfrak{m} : ~~~[y,x]_\mathfrak{h}= \big[y,x_\mathfrak{h}\big],~~~ \big[y,x\big]_\mathfrak{m}=
\big[y,x_\mathfrak{m}\big],~~~\mathrm{and}~~~ [z,x]_\mathfrak{h}= \big[z,x_\mathfrak{m}\big]_\mathfrak{h} .\tag{2}\] For any morphism \(\phi: (\mathfrak{g},\mathfrak{h},\mathfrak{m})\to
(\mathfrak{g}',\mathfrak{h}',\mathfrak{m}')\) we split the identity for the morphism into components: \[\label{EqCompMMComponentsMorph}
\forall~z_1,z_2\in\mathfrak{m}:~~ \phi_\mathfrak{h} \left([z_1,z_2]_\mathfrak{h}\right) =\left[\phi_\mathfrak{m}(z_1),\phi_\mathfrak{m}(z_1)\right]_{\mathfrak{h}'} ~~\mathrm{and}~~ \phi_\mathfrak{m} \left([z_1,z_2]_\mathfrak{m}\right)
=\left[\phi_\mathfrak{m}(z_1),\phi_\mathfrak{m}(z_1)\right]_{\mathfrak{m}'} .\tag{3}\]
Moreover, call a subalgebra \(\mathfrak{g}_1\subset \mathfrak{g}\) –where \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) is an RLA pair– transitive iff \(\mathfrak{m}\subset \mathfrak{g}_1\). Obviously, a transitive Lie subalgebra \(\mathfrak{g}_1\) is an RLA pair \((\mathfrak{g}_1,\mathfrak{h}\cap \mathfrak{g}_1, \mathfrak{m})\). As in [2], define \[\label{EqDefLieIdealI} \mathfrak{i}(\mathfrak{g},\mathfrak{h},\mathfrak{m}) := \mathfrak{i}(\mathfrak{g}):= \mathfrak{m} + [\mathfrak{m},\mathfrak{m}],\tag{4}\] and we call the RLA pair \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) \(\mathfrak{m}\)-generated iff \(\mathfrak{g}=\mathfrak{i}(\mathfrak{g})\). We have the following easy
Proposition 1. With the above notation: \(\mathfrak{i}(\mathfrak{g})\) is an ideal of \(\mathfrak{g}\) and equal to the minimal transitive subalgebra of \(\mathfrak{g}\) w.r.t. \(\mathfrak{m}\). Moreover, the subclass \(\mathrm{m}\mathcal{RLP}\) of all \(\mathfrak{m}\)-generated* RLA pairs is a full subcategory of \(\mathcal{RLP}\), and there is the following adjunction of functors \[\begin{tikzcd} \mathrm{m}\mathcal{RLP} \arrow[r, yshift=0.7ex, "\mathbf{J}"] &\mathcal{RLP}\arrow[l, yshift=-0.7ex, "\boldsymbol{\mathfrak{i}}"] \end{tikzcd}\] where \(\mathbf{J}\) is the inclusion functor and \(\boldsymbol{\mathfrak{i}}\) is the functor assigning to \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) the ideal \(\mathfrak{i}(\mathfrak{g})\). The unit of the adjunction is an isomorphism and the counit is the injection \(\mathfrak{i}(\mathfrak{g})\to \mathfrak{g}\) whence \(m\mathcal{RLP}\) is thus a coreflective subcategory of \(\mathcal{RLP}\), see [11].*
Proof. It is a routine check upon using \(\mathfrak{h}\)- and \(\mathfrak{m}\)-components that \(\mathfrak{i}(\mathfrak{g})\) is an ideal of \(\mathfrak{g}\) which obviously is transitive. Since every transitive subalgebra contains \(\mathfrak{m}\) and \([\mathfrak{m},\mathfrak{m}]\), the ideal \(\mathfrak{i}(\mathfrak{g})\) is contained in every transitive subalgebra, and thus the first statement is clear. Next, the fact that \[\label{EqCompHCompOfLieIdealI} \mathfrak{h}\cap \mathfrak{i}(\mathfrak{g}) =K\mathrm{span}\big\{[z_1,z_2]_\mathfrak{h} ~\big|~z_1,z_2\in\mathfrak{m}\big\}\tag{5}\] shows that \(\big(\mathfrak{i}(\mathfrak{g}),\mathfrak{i}(\mathfrak{g})\cap \mathfrak{h},\mathfrak{m}\big)\) is in \(\mathrm{m}\mathcal{RLP}\) and implies that each morphism \(\mathfrak{g}\to\mathfrak{g}'\) in \(\mathcal{RLP}\) restricts to a morphism of RLA pairs \(\mathfrak{i}(\mathfrak{g})\to \mathfrak{i}(\mathfrak{g}')\), see (3 ), whence \(\boldsymbol{\mathfrak{i}}\) is a well-defined functor. The adjunction properties are easily checked upon observing that for any \(\mathfrak{m}\)-generated RLA pair \(\mathfrak{g}\) and each morphism \(\mathfrak{g}\to\mathfrak{g}'\) of RLA pairs automatically corestricts to \(\mathfrak{i}(\mathfrak{g}')\subset \mathfrak{g}'\). ◻
Returning to an arbitrary RLA pair \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) one defines the following \(K\)-bilinear map \(T:\mathfrak{m}\times\mathfrak{m}\to \mathfrak{m}\) and \(K\)-trilinear map \(R:(\mathfrak{m}\times\mathfrak{m})\times \mathfrak{m}\to \mathfrak{m}\) by \[\begin{align} \forall~z,z'\in \mathfrak{m}:~~~ T(z,z') & := & -[z,z']_\mathfrak{m}, \tag{6}\\ \forall~z,z',z''\in \mathfrak{m}:~~~ R(z,z',z'')=:R(z,z')z'' & := & -\big[[z,z']_\mathfrak{h},z''\big], \tag{7} \end{align}\] compare [2]: this explains the unusual notation for the pair of brackets \((T,R)\) because of the relation to the torsion tensor \(T\) and the curvature tensor \(R\) of a particular canonical connection in the differential geometry of reductive homogeneous spaces, see e.g. [2].
Using the following notation for cyclic sums borrowed from [5]: given any \(K\)-modules \(V,W\) and any trilinear map \(\Xi:V\times V\times V\to W\) we write for all \(a,b,c\in V\) \[\forall~a,b,c\in V:~~~\mathfrak{S}_{(a,b,c)}\big(\Xi(a,b,c)\big):= \Xi(a,b,c)+\Xi(b,c,a)+\Xi(c,a,b).\] We obtain the following set of identities for \(T\) and \(R\) which can easily be deduced from the definitions (6 ) and (7 ) by taking certain \(\mathfrak{h}\)- and \(\mathfrak{m}\)-components in the Jacobi identity for the Lie bracket of \(\mathfrak{g}\) and from (3 ):
Proposition 2. Let \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) be an arbitrary RLA pair. Then for all \(z_1,z_2, z_3,z_4,z\in\mathfrak{m}\) the maps \(T\) and \(R\) satisfy the following six identities: \[\begin{align} T(z_1,z_2) & = & -T(z_2,z_1), \label{EqDefRLAToLieYAntisymT}\\ R(z_1,z_2)z & = & -R(z_2,z_1)z, \label{EqDefRLAToLieYAntisymR}\\ \mathfrak{S}_{(z_1,z_2,z_3)} \left(R(z_1,z_2)z_3-T\big(T(z_1,z_2),z_3\big)\right) & = & 0, \label{EqDefRLAToLieYCyclicRandTTBianchi1}\\ \mathfrak{S}_{(z_1,z_2,z_3)} \left(R\big(T(z_1,z_2),z_3\big)z\right) & = & 0,\label{EqDefRLAToLieYCyclicRT}\\ R(z_1,z_2)\big(T(z_3,z_4)\big)& = & T\big(R(z_1,z_2)z_3,z_4\big)+T\big(z_3,R(z_1,z_2)z_4\big), \label{EqDefRLAToLieYDeriROnT}\\ R(z_1,z_2)\big(R(z_3,z_4)z\big) & = & R\big(R(z_1,z_2)z_3,z_4\big)z +R\big(z_3,R(z_1,z_2)z_4\big)z\nonumber \\ & & +R(z_3,z_4)\big(R(z_1,z_2)z\big). \label{EqDefRLAToLieYDeriROnR} \end{align}\] {#eq: sublabel=eq:EqDefRLAToLieYAntisymT,eq:EqDefRLAToLieYAntisymR,eq:EqDefRLAToLieYCyclicRandTTBianchi1,eq:EqDefRLAToLieYCyclicRT,eq:EqDefRLAToLieYDeriROnT,eq:EqDefRLAToLieYDeriROnR} Moreover, let \(\phi:(\mathfrak{g},\mathfrak{h},\mathfrak{m})\to (\mathfrak{g}',\mathfrak{h}',\mathfrak{m}')\) be a morphism of RLA pairs. Then its \(\mathfrak{m}\)-component \(\phi_\mathfrak{m}:\mathfrak{m}\to \mathfrak{m}'\) maps \(T\) and \(R\) to their corresponding maps \(T'\) and \(R'\), respectively, i.e., \[\begin{align} \forall~z_1,z_2\in \mathfrak{m}:~~~\phi_\mathfrak{m}\big(T(z_1,z_2)\big) & = & T'\big(\phi_\mathfrak{m}(z_1), \phi_\mathfrak{m}(z_2)\big), \label{EqCompPhiSubMIntertwTTPrime}\\ \forall~z, z_1,z_2\in \mathfrak{m}:~~~\phi_\mathfrak{m}\big(R(z_1,z_2)z\big) & = & R'\big(\phi_\mathfrak{m}(z_1), \phi_\mathfrak{m}(z_2)\big)(\phi_\mathfrak{m}(z)). \label{EqCompPhiSubMIntertwRRPrime} \end{align}\] {#eq: sublabel=eq:EqCompPhiSubMIntertwTTPrime,eq:EqCompPhiSubMIntertwRRPrime}
Note that the identities (?? ) – (?? ) are a particular case of the Bianchi identities and some consequences for covariantly constant torsion and curvature, see e.g. [5]
The preceding results give rise to the following definition which is due to Kiyosi Yamaguti [3]:
Definition 1. Let \(E\) be a \(K\)-module equipped with a \(K\)-bilinear map \(T:E\times E\to E\) and a
\(K\)-trilinear map \(R:(E\times E)\times E\to E\) which satisfy the six identities (?? ) – (?? ). Then the pair \((E,T,R)\) is called a Lie-Yamaguti
algebra (short: LY algebra).
Let \((E',T',R')\) be another Lie-Yamaguti algebra. A \(K\)-linear map \(\psi:E\to E'\) is called a morphism of Lie-Yamaguti algebras if for
all \(z_1,z_2,z_3\in E\) \[\label{EqDefMorphLY} \psi\big(T(z_1,z_2)\big) = T'\big(\psi(z_1), \psi(z_2)\big)~~\mathrm{and}~~ \psi\big(R(z_1,z_2)z_3\big)
= R'\big(\psi(z_1), \psi(z_2)\big)(\psi(z_3)).\tag{8}\] Hence \(\psi\) intertwines the corresponding maps \(T,R,T',R'\) as \(\phi_\mathfrak{m}\) does in (?? ) and (?? ).
K. Yamaguti has denoted the map \((v,w,z)\mapsto -R(v,w)z\) by a triple bracket \((v,w,z)\mapsto[v,w,z]\) and the map \((v,w)\mapsto -T(v,w)\) by a
multiplication \((v,w)\mapsto v\circ w\), and he has called these objects ‘Lie triple algebras’. The motivating particular case \(T=0\) reduces the six conditions of Proposition 2 to (?? ), (?? ), and (?? ) in which case the Lie-Yamaguti algebra is called a Lie triple system, see e.g. [4]. The other extreme case \(R=0\) reduces to (?? ) and (?? ) which means that \((E,T)\) is a Lie algebra.
It follows at once that Lie-Yamaguti algebras together with their morphisms form a category \(\mathcal{LY}\), and that there is an obvious functor \(\mathbf{G}': \mathcal{LY} \longleftarrow
\mathcal{RLP}\) assigning to each RLA pair \((\mathfrak{g},\mathfrak{h},\mathfrak{m})\) the LY algebra \((\mathfrak{m},T,R)\) according to (6 ) and (7 ), and to each RLA pair morphism \(\phi\) the component \(\phi_\mathfrak{m}\). It is easy to see that \(\mathbf{G}'\)
factorizes as \(\mathbf{G}'=\mathbf{G}\boldsymbol{\mathfrak{i}}\): \[\label{EqDefFunctorG} \begin{tikzcd} \mathcal{LY} & \mathrm{m}\mathcal{LRP}
\arrow[l, "\mathbf{G}"'] & \mathcal{LRP} \arrow[l, "\boldsymbol{\mathfrak{i}}"'] \end{tikzcd}\tag{9}\] where \(\mathbf{G}\) is the ‘restriction’ of \(\mathbf{G}'\) to \(\mathrm{m}\mathcal{LRP}\).
There is a well-established assignment of a RLA pair \(\big(\hat{\mathfrak{g}}(E), \hat{\mathfrak{h}}(E),E\big)\) to every Lie-Yamaguti algebra \((E,T,R)\) –which is called its
enveloping Lie algebra– in the following way, see e.g. [3]: consider the \(K\)-submodule \(\hat{\mathfrak{h}}(E)\) of \(\mathrm{Hom}_K(E,E)\) which is spanned by all the \(K\)-linear maps of the form \(\hat{R}(v_1,v_2):z\mapsto R(v_1,v_2)z\), i.e., \[\label{EqDefHatHOfE}
\hat{\mathfrak{h}}(E):=K\mathrm{span} \big\{ -\hat{R}(v_1,v_2)\in\mathrm{Hom}_K(E,E)~\big|~ v_1,v_2\in E\big\}.\tag{10}\] Thanks to (?? ) the \(K\)-submodule \(\hat{\mathfrak{h}}(E)\), equipped with the commutator of linear maps, is a Lie subalgebra of \(\mathrm{Hom}_K(E,E)\). Define \[\label{EqDefEnvelLieAlg} \hat{\mathfrak{g}}(E):= \hat{\mathfrak{h}}(E)\oplus E\tag{11}\] with the bracket \[\label{EqDefEnvelLieAlgBracket}
\forall~\xi,\eta\in \hat{\mathfrak{h}}(E)~\forall~v,w\in E: [\xi+v,\eta+w]^\wedge:=\big(\xi\circ \eta-\eta\circ\xi-\hat{R}(v,w)\big) ~~+~~\big(\xi(w)-\eta(v)-T(v,w)\big).\tag{12}\] Since the 1950’s it is well known and not hard to check by
using the Lie-Yamaguti identities that \(\big(\hat{\mathfrak{g}}(E),\hat{\mathfrak{h}}(E),E\big)\) is a reductive Lie algebra pair.
Although the enveloping Lie algebra has turned out to be very useful in differential geometry, the algebraic draw-back is the fact that the assignment of a Lie-Yamaguti algebra to its enveloping Lie algebra cannot be defined on morphisms of Lie-Yamaguti
algebras in a functorial way: it follows that this assignment does not lead to a functor \(\mathcal{LY}\to\mathcal{RLP}\) 4
as the following counter-example shows:
Example 1. We shall construct two Lie triple systems, hence particular LY algebras with vanishing binary bracket, by means of the following well known construction: Recall that for an arbitrary Lie algebra \(\big(\mathfrak{l},[~,~]\big)\) there is always a Lie triple system \(\big(E(\mathfrak{l})=\mathfrak{l},R_\mathfrak{l}\big)\) where the ternary bracket \(R_\mathfrak{l}\) is defined by \[\forall~x,y,z\in\mathfrak{l}:~~~R_\mathfrak{l}(x,y)z:=-\big[[x,y],z\big].\] Furthermore, it is easy to check that the assignment of \(\big(\mathfrak{l},[~,~]\big)\) to \(\big(E(\mathfrak{l}),R_\mathfrak{l}\big)\) defines an obvious functor from the category of all Lie algebras to the category of all Lie triple systems. For
each \(x\in\mathfrak{l}\) let \(\mathrm{ad}_x:\mathfrak{l}\to \mathfrak{l}\) denote the usual adjoint representation \(z\mapsto \mathrm{ad}_x(z)=[x,z]\). It
is not hard to see from the above definition of \(R_\mathfrak{l}\) that the enveloping Lie algebra \(\hat{\mathfrak{g}}(E(\mathfrak{l}))\) of \(\big(E(\mathfrak{l}),R_\mathfrak{l}\big)\) is given by the RLA pair \(\big(\mathrm{ad}_{[\mathfrak{l},\mathfrak{l}]}\oplus
\mathfrak{l},\mathrm{ad}_{[\mathfrak{l},\mathfrak{l}]},\mathfrak{l}\big)\) where \(\mathrm{ad}_{[\mathfrak{l},\mathfrak{l}]}\subset \mathrm{Hom}_\mathbb{K}(\mathfrak{l},\mathfrak{l})\) denotes the Lie subalgebra of
the Lie algebra \(\mathrm{Hom}_\mathbb{K}(\mathfrak{l},\mathfrak{l})\) spanned by all linear maps of the form \(\mathrm{ad}_{[x,y]}\) for some \(x,y\in\mathfrak{l}\). According to (12 ) the Lie bracket on \(\hat{\mathfrak{g}}(E(\mathfrak{l}))\) is given by \[\forall~\xi,\eta\in \mathrm{ad}_{[\mathfrak{l},\mathfrak{l}]}~ \forall~x,y\in\mathfrak{l}:~~ \big[\xi+x,\eta+y\big]^\wedge= \big(\xi\circ \eta-\eta\circ\xi+\mathrm{ad}_{[x,y]}\big) ~+~\big(\xi(y)-\eta(x)\big).\] Consider a
field \(\mathbb{K}\) of characteristic \(0\), let \(\mathfrak{l}\) be the abelian Lie algebra \(\mathbb{K}^2\), and let
\(\mathfrak{l}'\) be the six-dimensional nilpotent Lie algebra of all strictly upper triangular \(4\times 4\) matrices. Let \(e_1,e_2\) be the canonical
basis of \(\mathbb{K}^2\). Denoting by \(E_{ij}\) the standard elementary \(4\times 4\)-matrices for all \(1\leq i,j\leq 4\)
it is immediate that the six matrices \(E_{12},E_{13}, E_{14}, E_{23}, E_{24}, E_{34}\) form a basis for the Lie algebra \(\mathfrak{l}'\). Recall the equations for the Lie brackets
\[\forall~1\leq i,j,k,l\leq 4:~~~ [E_{ij},E_{kl}] =\delta_{jk}E_{il}-\delta_{il}E_{kj}\] with the usual Kronecker delta \(\delta_{ij}=1\) if \(i=j\) and
\(\delta_{ij}=0\) if \(i\not = j\). We observe that the \(4\)-dimensional subspace \(I\subset \mathfrak{l}'\) spanned by
\(E_{13},E_{14},E_{24}, E_{34}\) forms an ideal of \(\mathfrak{l}'\), and that the linear map \(p:\mathfrak{l}'\to\mathfrak{l}\) sending \(E_{12}\) to \(e_1\) and \(E_{23}\) to \(e_2\) and \(I\) to \(\{0\}\) is a surjective morphism of Lie algebras. Hence, we get a morphism –also denoted by \(p\)– of the Lie triple system \(\big(E(\mathfrak{l}'),R_{\mathfrak{l}'}\big)\) to the abelian Lie triple system \(\big(E(\mathfrak{l}),0\big)\). On the other hand, consider the linear map \(i:\mathfrak{l}\to \mathfrak{l}'\) which sends \(e_1\) to \(E_{12}\) and \(e_2\) to \(E_{23}\). This is not a morphism of Lie algebras because \([e_1,e_2]=0\), but \([E_{12},E_{23}]=E_{13}\not = 0\). However, the map \(i\) is a morphism of the associated Lie triple systems since \[R_{\mathfrak{l}'}\big(i(e_1),i(e_2)\big)(i(e_1))= R_{\mathfrak{l}'}(E_{12},E_{23})E_{12}=-\big[[E_{12},E_{23}],E_{12}\big]
=-[E_{13},E_{12}]=0=i\big(R_\mathfrak{l}(e_1,e_2)e_1\big)\] and likewise \(R_{\mathfrak{l}'}(E_{12},E_{23})E_{23}=0\). It follows that we get the following diagram of Lie triple systems \[\begin{tikzcd} E(\mathfrak{l}) \arrow[r, yshift=0.7ex, "i"] & E(\mathfrak{l}') \arrow[l, yshift=-0.7ex, "p"] ~~~\mathrm{with}~~~ p\circ i=\mathrm{id}_{E(\mathfrak{l})}.
\end{tikzcd}\] Since \(\mathfrak{l}\) is abelian, we have \(\mathrm{ad}_{\mathfrak{l}}=\{0\}\). Hence, the enveloping Lie algebra \(\hat{\mathfrak{g}}(E(\mathfrak{l}))\) is isomorphic to the RLA pair \(\big(\mathfrak{l},0,\mathfrak{l}\), hence, to the abelian Lie algebra \(\mathfrak{l}\).
Moreover, the enveloping Lie algebra \(\hat{\mathfrak{g}}(E(\mathfrak{l}'))\) is easily computed to be isomorphic to the RLA pair \[\hat{\mathfrak{g}}(E(\mathfrak{l}')) \cong
\left(\mathbb{K}\mathrm{span}\left\{\mathrm{ad}_{E_{13}}, \mathrm{ad}_{E_{24}}\right\} \oplus \mathfrak{l}', \mathbb{K}\mathrm{span}\left\{\mathrm{ad}_{E_{13}}, \mathrm{ad}_{E_{24}}\right\}, \mathfrak{l}'\right).\] Note that both linear maps
\(\mathrm{ad}_{E_{13}}\) and \(\mathrm{ad}_{E_{24}}\) vanish on \(E_{23}, E_{13}, E_{24}\), and \(E_{14}\), that \(\mathrm{ad}_{E_{13}}\) vanishes on \(E_{12}\) and sends \(E_{34}\) to the central element \(E_{14}\), whereas \(\mathrm{ad}_{E_{24}}\) vanishes on \(E_{34}\) and sends \(E_{12}\) to the central element \(-E_{14}\). It follows that the
linear maps \(\mathrm{ad}_{E_{13}}\) and \(\mathrm{ad}_{E_{24}}\) are linearly independent.
Suppose there was a functor \(\mathbf{H}:\mathcal{LY}\to \mathcal{RLP}\) assigning to each Lie-Yamaguti algebra \(E\) its enveloping Lie algebra \(\mathbf{H}E:=\hat{\mathfrak{g}}(E)\). Applying the functor \(\mathbf{H}\) to the above diagram we would get a corresponding diagram \[\begin{tikzcd}
\hat{\mathfrak{g}}(E(\mathfrak{l})) \arrow[r, yshift=0.7ex, "\mathbf{H}i"] & \hat{\mathfrak{g}}(E(\mathfrak{l}')) \arrow[l, yshift=-0.7ex, "\mathbf{H}p"] ~~~\mathrm{with}~~~ \mathbf{H}p\circ \mathbf{H}i
=\mathbf{H}\mathrm{id}_{E(\mathfrak{l})}= \mathrm{id}_{\hat{\mathfrak{g}}(E(\mathfrak{l}))}.
\end{tikzcd}\] It follows that the morphism of RLA pairs \(\mathbf{H}i\) would be injective whereas the morphism of RLA pairs \(\mathbf{H}p\) would be surjective: this implies that
the image \(\mathbf{H}i\big(\hat{\mathfrak{g}}(E(\mathfrak{l}))\big)\) would be a two-dimensional abelian subalgebra of the \(\mathfrak{l}'\)-part of \(\hat{\mathfrak{g}}(E(\mathfrak{l}'))\) which would be a vector space complement to the subspace \(I\subset \mathfrak{l}'\). A basis of \(\mathbf{H}i\big(\hat{\mathfrak{g}}(E(\mathfrak{l}))\big)\) could then be given by the matrices \(E_{12}+A\) and \(E_{23}+B\) with some \(A,B\in I\). An easy computation shows that there would be \(\lambda\in\mathbb{K}\) such that \[\big[E_{12}+A,E_{23}+B\big]^\wedge=
\mathrm{ad}_{[E_{12}+A,E_{23}+B]}=\mathrm{ad}_{E_{13}}+\lambda \mathrm{ad}_{E_{24}}\not =0\] thanks to the linear independence of \(\mathrm{ad}_{E_{13}}\) and \(\mathrm{ad}_{E_{24}}\). But this is in contradiction to the fact that the subspace \(\mathbf{H}i\big(\hat{\mathfrak{g}}(E(\mathfrak{l}))\big)\) is abelian. Therefore such a functor \(\mathbf{H}\) cannot exist. \(\Box\)
In this section we should like to construct a left adjoint functor \[\mathbf{F}:\mathcal{LY} \longrightarrow \mathrm{m}\mathcal{RLP}\] to the functor \(\mathbf{G}\), see (9 ) which will give a sort of ‘free object in \(\mathrm{m}\mathcal{RLP}\) generated by a given Lie-Yamaguti algebra’. A similar construction has already been done for the more particular Lie
triple systems, see [10].
In order to do so, we start with a Lie-Yamaguti algebra \((E,T,R)\), and we form the \(K\)-modules \[\label{EqDefTildeGE}
\tilde{\mathfrak{m}}(E) := E,~~~~ \tilde{\mathfrak{h}}(E) :=\Lambda^2E,~~~~\tilde{\mathfrak{g}}(E):= E\oplus \Lambda^2E= \tilde{\mathfrak{m}}_E\oplus \tilde{\mathfrak{h}}(E).\tag{13}\] We define the following \(K\)-bilinear multiplication \([~,~]^\sim:\tilde{\mathfrak{g}}(E)\times \tilde{\mathfrak{g}}(E)\to \tilde{\mathfrak{g}}(E)\) for all \(v,v_1,v_2\), \(w,w_1,w_2\in E\) \[\begin{align} [v_1,v_2]^\sim & := & -T(v_1,v_2)+ v_1\wedge v_2, \tag{14} \\ ~[w_1\wedge w_2, v]^\sim & := & -R(w_1,w_2)v =: -[v,w_1\wedge
w_2]^\sim, \tag{15} \\ ~[v_1\wedge v_2,w_1\wedge w_2]^\sim & := & -\big(R(v_1,v_2)w_1\big)\wedge w_2 -w_1\wedge \big(R(v_1,v_2)w_2\big). \tag{16}
\end{align}\] We shall see in Lemma 3 that this bracket is well-defined whence the pair \(\big(\tilde{\mathfrak{g}},[~,~]^\sim\big)\) will be a non-associative
algebra. However, it will turn out that the above bracket \(\big[~,~\big]^\sim\) is not a Lie bracket, it is a priori not even antisymmetric, see (16 ). In order to get an idea for
the modification of this bracket \([~,~]^\sim\) on a factor module, it has turned out to be convenient to use the theory of the slightly more general (left) Leibniz algebras, see e.g. [12] for right Leibniz algebras and e.g. [13] for left Leibniz algebras: for a given \(K\)-module \(V\) and \(K\)-bilinear map \([~,~]^\vee:V\times V\to V\) we set:
\[\label{EqDefLeibnizIdentityMapL} \forall~\xi_1,\xi_2,\xi_3\in V:~~~ \mathbf{L}(\xi_1,\xi_2,\xi_3):= \big[\xi_1,[\xi_2,\xi_3]^\vee\big]^\vee
-\big[[\xi_1,\xi_2]^\vee,\xi_3\big]^\vee -\big[\xi_2,[\xi_1,\xi_3]^\vee\big]^\vee.\tag{17}\] The pair \((V,[~,~]^\vee)\) is called a Leibniz algebra iff \(\mathbf{L}(\xi_1,\xi_2,\xi_3)=0\) for all \(\xi_1,\xi_2,\xi_3\in V\). Recall that every Leibniz algebra whose bracket is antisymmetric is a Lie algebra. Moreover, recall that for every Leibniz
algebra \(\big(V,[~,~]^\vee\big)\) the \(K\)-submodule \[\label{EqDefLeibnizQuadraticIdeal} q(V)
:=K\mathrm{span}\big\{[v,w]^\vee+[w,v]^\vee~|~ v,w\in V\big\}\tag{18}\] is a two-sided abelian ideal of \(\big(V,[~,~]^\vee\big)\)– meaning that the induced multiplication vanishes–, and that the quotient
Leibniz algebra \(\overline{V}:=V/q(V)\) is a Lie algebra. We first need the following
Lemma 3. Let \((E,T,R)\) be a Lie-Yamaguti algebra. Then the bilinear bracket \([~,~]^\sim\) on \(\tilde{\mathfrak{g}}(E)\) given by eqs (14 ), (15 ), and (16 ) is well-defined and satisfies the following six equations for all \(v,v_1,v_2,w,w_1,w_2,z,z_1,z_2\in E\) \[\begin{align} \mathbf{L}(v_1,v_2,z)& = & \mathfrak{S}_{(v_1,v_2,z)} \left(T(v_1,v_2)\wedge z\right)~\in~ \tilde{\mathfrak{h}}(E), \tag{19} \\ \mathbf{L}(v,w_1\wedge w_2, z) &= & -~\mathbf{L}(w_1\wedge w_2,v,z)~=~0, \tag{20}\\ \mathbf{L}(v_1\wedge v_2,w_1\wedge w_2, z) & = & 0, \tag{21}\\ \mathbf{L}(v_1, v_2,w_1\wedge w_2) & = & \big(R(v_1,v_2)w_1\big)\wedge w_2 +w_1\wedge \big(R(v_1,v_2)w_2\big)\nonumber \\ & & +\big(R(w_1,w_2)v_1\big)\wedge v_2 +v_1\wedge \big(R(w_1,w_2)v_2\big)\nonumber \\ & = & -[v_1\wedge v_2,w_1\wedge w_2]^\sim -[w_1\wedge w_2,v_1\wedge v_2]^\sim ~\in \tilde{\mathfrak{h}}(E), \tag{22} \\ \mathbf{L}(z,v_1\wedge v_2,w_1\wedge w_2) & = & -\mathbf{L}(v_1\wedge v_2,z,w_1\wedge w_2)~=~0, \tag{23}\\ \mathbf{L}(v_1\wedge v_2,w_1\wedge w_2,z_1\wedge z_2) & = & 0. \tag{24} \end{align}\]
Proof. The bracket \([~,~]^\sim\) is well-defined thanks to the antisymmetry of \(T\) and \(R\) in the left two arguments and to the fact that the exterior algebra \(\Lambda E\) is the free graded commutative algebra generated by \(E\). The six identities are straight-forward computations: (19 ) follows from identity (?? ), (20 ) follows from identity (?? ), (21 ) follows from identity (?? ), (22 ) follows from identity (?? ), ( 23 ) follows from identity (?? ), and (24 ) is deduced from identity (?? ). ◻
Now the definition of the bracket \([~,~]^\sim\) in eqs (14 ), (15 ) and (16 ), and statement (24 ) of Lemma 3 implies the obvious
Corollary 1. With the hypotheses of the preceding Lemma 3 we have
\(\tilde{\mathfrak{h}}(E)\) is a subalgebra of \(\tilde{\mathfrak{g}}(E)\) and satisfies the Leibniz identity.
We have \[\big[\tilde{\mathfrak{h}}(E), \tilde{\mathfrak{m}}_E\big]^\sim \subset \tilde{\mathfrak{m}}_E ~~~\mathrm{and}~~~ \big[\tilde{\mathfrak{m}}_E, \tilde{\mathfrak{h}}(E)\big]^\sim \subset \tilde{\mathfrak{m}}_E\] and identities (21 ) and (23 ) imply that \(\tilde{\mathfrak{m}}_E=E\) is a so-called symmetric Leibniz bimodule of \(\tilde{\mathfrak{h}}(E)\), see Section 3 in [13].
Next, it is reasonable to define the following \(K\)-submodules of \(\tilde{\mathfrak{h}}(E)\subset \tilde{\mathfrak{g}}(E)\) which describe the failure of the Leibniz identity of \(\big(\tilde{\mathfrak{g}}(E),[~,~]^\sim\big)\): \[\begin{align} \tilde{I}_1 & := & K\mathrm{Span}\Big\{ \big[v_1\wedge v_2, w_1\wedge w_2\big]^\sim +\big[w_1\wedge w_2, v_1\wedge v_2\big]^\sim~ \Big|~v_1,v_2,w_1,w_2\in E \Big\}, \tag{25}\\ \tilde{I}_2 & := & K\mathrm{Span}\left\{ \mathfrak{S}_{(v,v_1,v_2)}\big(v\wedge T(v_1,v_2)\big)~ \big|~v,v_1,v_2\in E \right\}, \tag{26}\\ \tilde{I}(E) & := & \tilde{I}_1 +\tilde{I}_2. \tag{27} \end{align}\]
Lemma 4. With the notation of the preceding Lemma 3:
\(\tilde{I}_1\) and \(\tilde{I}_2\), and hence, \(\tilde{I}(E)\) are two-sided ideals of \(\tilde{\mathfrak{g}}\) contained in the subalgebra \(\tilde{\mathfrak{h}}\).
The factor algebra \[\label{EqDefGE} \mathfrak{g}(E):= \tilde{\mathfrak{g}}(E)/\tilde{I}(E)\tag{28}\] gives rise to a \(\mathfrak{m}\)-generated reductive Lie algebra pair \(\big(\mathfrak{g}(E),\mathfrak{h}(E),E\big)\) where \[\label{EqDefHE} \mathfrak{h}(E):= \tilde{\mathfrak{h}}(E)/\tilde{I}(E).\tag{29}\]
Let \((\mathfrak{g}',\mathfrak{h}',\mathfrak{m}')\) be a RLA pair with Lie bracket \([~,~]'\). Let \(\chi:(E,T,R)\to (\mathfrak{m}',T',R')\) be a morphism of Lie-Yamaguti algebras, where \(T'\) and \(R'\) are defined as in (6 ) and (7 ). Then the \(K\)-linear map \(\tilde{\chi}:\tilde{\mathfrak{g}}(E)\to \mathfrak{g}'\) given by \[\label{EqDefFofPsi} \forall~v,v_1,v_2\in E:~~~~~ \tilde{\chi}(v):=\chi(v)~~~\mathrm{and}~~~ \tilde{\chi}\big(v_1\wedge v_2 \big):= \big[\chi(v_1),\chi(v_2)\big]'_{\mathfrak{h}'}\tag{30}\] is a morphism of nonassociative algebras and passes to the quotient to define a morphism of RLA pairs \(\check{\chi}:\mathfrak{g}(E)\to \mathfrak{g}'\).
Proof. i.) According to Corollary 1, the submodule \(\Lambda^2E=\tilde{\mathfrak{h}}(E)\) is a subalgebra of \(\big(\tilde{\mathfrak{g}}(E),[~,~]^\sim\big)\) and satisfies the Leibniz identity. Since the \(K\)-submodule \(I_1\) obviously coincides with the ideal of
squares \(q\big(\tilde{\mathfrak{h}}(E)\big)\), see eqn (18 ), it is an abelian two-sided ideal of the subalgebra \(\tilde{\mathfrak{h}}(E)\),
i.e. \(\big[I_1,\tilde{\mathfrak{h}}(E)\big]^\sim=\{0\}\) and \(\big[\tilde{\mathfrak{h}}(E),I_1\big]^\sim\subset I_1\). Next, by means of identity (21 )
it can be shown by an easy computation that \(\{0\}=\big[\tilde{I}_1,E\big]^\sim \stackrel{(\ref{EqDefTildeBracketEW2E})}{=} -\big[E,\tilde{I}_1\big]^\sim\) which proves that \(\tilde{I}_1\)
is a two-sided ideal of \(\big(\tilde{\mathfrak{g}}(E),[~,~]^\sim\big)\).
Secondly, for the \(K\)-submodule \(\tilde{I}_2\) identity (?? ) yields \(\{0\}=\big[\tilde{I}_2,E\big]^\sim \stackrel{(\ref{EqDefTildeBracketEW2E})}{=}
-\big[E,\tilde{I}_2\big]^\sim\). Moreover identities (16 ) and (?? ) imply \(\big[\tilde{I}_2,\tilde{\mathfrak{h}}(E)\big]= \{0\}\). On the other hand, thanks to the
definition of the bracket (16 ) and to identity (?? ) we get after a longer computation \(\big[\tilde{\mathfrak{h}}(E),\tilde{I}_2\big] \subset \tilde{I}_2\) whence \(\tilde{I}_2\) is also a two-sided ideal of \(\big(\tilde{\mathfrak{g}}(E),[~,~]^\sim\big)\). Finally, the sum of two two-sided ideals is always a two-sided ideal. Hence, \(\tilde{I}(E)\) also is a two-sided ideal.
ii.) Let \(\varpi: \tilde{\mathfrak{g}}(E)\to \mathfrak{g}(E) =\tilde{\mathfrak{g}}(E)/\tilde{I}(E)\) be the canonical projection which is a morphism of non-associative algebras. Hence, for all \(\xi,\eta,\zeta\in \tilde{\mathfrak{g}}(E)\) it sends each term of the form \(\mathbf{L}(\xi,\eta,\zeta)\big)\), see (17 ), to the corresponding term
\(\mathbf{L}\big(\varpi(\xi),\varpi(\eta),\varpi(\zeta)\big)\) in the factor algebra \(\mathfrak{g}(E)\). On the other hand, by the definition (27
) of \(\tilde{I}(E)\) the term \(\mathbf{L}(\xi,\eta,\zeta)\big)\) belongs to \(\tilde{I}(E)= \mathrm{Ker}(\varpi)\) thanks to the statements of Lemma 3. This implies that \(\mathbf{L}\big(\varpi(\xi),\varpi(\eta),\varpi(\zeta)\big)=0\), and the factor algebra \(\mathfrak{g}(E)\) thus satisfies the Leibniz identity. Moreover, the bracket \([~,~]\) on \(\mathfrak{g}(E)\) induced by \([~,~]^\sim\) is antisymmetric (since \(\tilde{I}_1\subset\tilde{I}(E)\)) whence the bracket on \(\mathfrak{g}(E)\) is a Lie bracket. Obviously, since \(\tilde{I}(E)\subset \tilde{\mathfrak{h}}(E)\) we can infer that \(\mathfrak{h}(E)=\tilde{\mathfrak{h}}(E)/\tilde{I}(E)\) is a subalgebra of \(\mathfrak{g}(E)\),
and since \(\tilde{I}(E)\cap E=\{0\}\) we have that the image of the subspace \(E\) modulo \(\tilde{I}(E)\) is isomorphic to \(E/(\tilde{I}(E)\cap E)=E/\{0\}\cong E\). Finally, the definition of the bracket in (15 ) shows that \(E\) is invariant under the adjoint action of \(\mathfrak{h}(E)\), and therefore \((\mathfrak{g}(E),\mathfrak{h}(E),E)\) is a RLA pair.
iii.) The map \(\tilde{\chi}\) is a well-defined \(K\)-linear map thanks to the universal properties of the Grassmann algebra \(\Lambda E\). The
fact that \(\tilde{\chi}\) is a morphism of nonassociative algebras \(\big(\tilde{\mathfrak{g}}(E),[~,~]^\sim\big) \to \big(\mathfrak{g}',[~,~]'\big)\), i.e. for all \(\xi_1,\xi_2\in \tilde{\mathfrak{g}}(E)\): \(\tilde{\chi}([\xi_1,\xi_2]^\sim) =[\tilde{\chi}(\xi_1),\tilde{\chi}(\xi_2)]'\), is a straight-forward computation using (14 ), (30 ), and (6 ) for any \(\xi_1=v_1,\xi_2=v_2\in E\), (15 ), (7 ), and (30 ) for any \(\xi_1=v\in E,\xi_2=v_1\wedge v_2\in \Lambda^2E\), and (15 ), (7 ),
(30 ), and the Jacobi identity for the Lie bracket \([~,~]'\) for all \(v_1\wedge v_2,w_1\wedge w_2\in\Lambda^2E\).
According to (25 ) the ideal \(\tilde{I}_1\) is spanned by \(\big[v_1\wedge v_2, w_1\wedge w_2\big]^\sim +\big[w_1\wedge w_2, v_1\wedge v_2\big]^\sim\). Using
the fact that \(\tilde{\chi}\) is a morphism of non-associative algebras and Definition (30 ) we get \[\tilde{\chi}\Big(\big[v_1\wedge v_2, w_1\wedge
w_2\big]^\sim\Big) =\big[\tilde{\chi}(v_1\wedge v_2),\tilde{\chi}(w_1\wedge w_2)\big]' =\Big[\big[\chi(v_1),\chi(v_2)\big]'_{\mathfrak{h}'}, \big[\chi(w_1),\chi(w_2)\big]'_{\mathfrak{h}'} \Big]'.\] The right hand side of this
equation is antisymmetric in the arguments \(v_1\wedge v_2\) and \(w_1\wedge w_2\) since \([~,~]'\) is a Lie bracket on \(\mathfrak{g}'\). This implies that \(\tilde{\chi}\) vanishes on \(\tilde{I}_1\). Moreover, the fact that \(\tilde{\chi}(\tilde{I}_2)=\{0\}\) is computed in a straight-forward manner using eqs (30 ), (6 ), and again the Jacobi identity for the Lie bracket \([~,~]'\). It follows that \(\tilde{\chi}\) maps the ideal \(\tilde{I}(E)=\tilde{I}_1+\tilde{I}_2\) of \(\tilde{\mathfrak{g}}_{E}\) to \(\{0\}\) whence the map \(\tilde{\chi}\) passes to the quotient to a well-defined morphism of Lie algebras \(\check{\chi}:\mathfrak{g}(E)\to \mathfrak{g}'\) mapping the subalgebra \(\mathfrak{h}(E)\) to the subalgebra \(\mathfrak{h}'\) and the \(K\)-submodule \(E\) to the \(K\)-submodule \(\mathfrak{m}'\). It follows that \(\check{\chi}\) is a morphism of RLA pairs. ◻
Corollary 2. The assignment \[\label{EqDefFunctorF} \mathbf{F}: \mathcal{LY} \longrightarrow \mathrm{m}\mathcal{RLP},\tag{31}\] which associates to each Lie-Yamaguti algebra \((E,T,R)\) the reductive Lie algebra pair \(\mathbf{F}(E,T,R)=(\mathfrak{g}(E),\mathfrak{h}(E),E)\), see (28 ) and (29 ), and to each morphism \(\psi:(E,T,R)\to (E',T',R')\) the morphism \(\mathbf{F}\psi=\check{\psi}: (\mathfrak{g}(E),\mathfrak{h}(E),E)\to (\mathfrak{g}(E'),\mathfrak{h}(E'),E')\), see (30 ), is a covariant functor.
Proof. The fact that \(\mathbf{F}(E,T,R)\) is a RLA pair is shown in statement ii.) of the preceding Lemma 4. Moreover, set the RLA pair \((\mathfrak{g}',\mathfrak{h}',\mathfrak{g}')\) occurring in statement iii.) of Lemma 4 equal to \((\mathfrak{g}(E'),\mathfrak{h}(E'),E')\). Since it is obvious that its associated LY algebra \(\mathbf{G}\mathfrak{g}(E')\) (with binary and ternary brackets defined by (6 and (7 )) is equal to the LY algebra \((E',T',R')\) it follows that the morphism \(\mathbf{F}\psi=\check{\psi}\) is well-defined thanks to statement \(iii)\) of lemma 4. The fact that \(\mathbf{F}\) preserves composition of morphisms and maps identity morphisms to identity morphisms is a straight-forward consequence of the preceding results and the definitions. Hence \(\mathbf{F}\) is a covariant functor. ◻
In the following theorem we shall show that the functor \(\mathbf{F}\) is a left adjoint of the more obvious functor \(\mathbf{G}\), see (9 ), and thus makes the RLA pair \((\mathfrak{g}(E),\mathfrak{h}(E),E)\) a universal object which may be called the free RLA pair generated by the LY algebra \((E,T,R)\):
Theorem 1. The functor \(\mathbf{F}\) defined in Corollary 2 is a left adjoint to the functor \(\mathbf{G}\), see (9 ). The natural isomorphism of the adjunction \[\nu_{E,\mathfrak{g}'}:\label{EqDefAdjugantKModules} \mathrm{Hom}_{\mathrm{m}\mathcal{RLP}}\big(\mathbf{F}(E,T,R), (\mathfrak{g}',\mathfrak{h}',\mathfrak{m}')\big) \to \mathrm{Hom}_\mathcal{LY} \big((E,T,R),\mathbf{G}(\mathfrak{g}',\mathfrak{h}',\mathfrak{m}')\big)\tag{32}\] is defined by the restriction \[\label{EqDefAdjugantElements} \nu_{E,\mathfrak{g}'}(\vartheta):=\vartheta|_{E}:E\to \mathfrak{m}'.\tag{33}\] Moreover, the components of the unit of the adjunction \(\boldsymbol{\eta}_E:E\to \mathbf{GF}(E)\) are natural isomorphisms of Lie-Yamaguti algebras, and the components of the counit of the adjunction \(\boldsymbol{\epsilon}_{\mathfrak{g}}:\mathbf{FG}(\mathfrak{g}) \to\mathfrak{g}\) are surjective natural morphisms of \(\mathfrak{m}\)-generated RLA pairs.
Proof. Since each morphism of RLA pairs \(\vartheta:\mathfrak{g}(E)\to \mathfrak{g}'\) maps \(E\) to the submodule \(\mathfrak{m}'\) of
\(\mathfrak{g}'\), the restriction is well-defined and a morphism of Lie-Yamaguti algebras. According to Proposition 2, it follows that
\(\nu_{E,\mathfrak{g}'}\) is well-defined morphism.
We shall first prove naturality of \(\nu\) in both of its arguments: given an arbitrary morphism \(\zeta:(E_1,T_1,R_1)\to (E_2,T_2,R_2)\) of Lie-Yamaguti algebras and an arbitrary morphism
\(\phi':\mathfrak{g}'_1\to \mathfrak{g}'_2\) of RLA pairs we have to prove the commutativity of the following two diagrams: \[\label{EqDiagrams}
\begin{tikzcd} \mathrm{Hom}_{\mathrm{m}\mathcal{RLA}} \big(\mathfrak{g}(E_1),\mathfrak{g}'\big) \arrow[r, "\nu_{E_1,\mathfrak{g}'}"] & \mathrm{Hom}_{\mathcal{LY}}(E_1,\mathfrak{m}')\\ \mathrm{Hom}_{\mathrm{m}\mathcal{RLA}}
\big(\mathfrak{g}(E_2),\mathfrak{g}'\big) \arrow[r, "\nu_{E_2,\mathfrak{g}'}"] \arrow[u, "(~)\circ F(\zeta)"] & \mathrm{Hom}_{\mathcal{LY}}(E_2,\mathfrak{m}') \arrow[u, "(~)\circ \zeta"'] \end{tikzcd}
~~\mathrm{and}~~ \begin{tikzcd} \mathrm{Hom}_{\mathrm{m}\mathcal{RLA}} \big(\mathfrak{g}(E),\mathfrak{g}'_1\big) \arrow[r, "\nu_{E,\mathfrak{g}_1'}"] \arrow[d, "\phi'\circ (~)"'] &
\mathrm{Hom}_{\mathcal{LY}}(E,\mathfrak{m}_1') \arrow[d, "G(\phi')\circ(~)"] \\ \mathrm{Hom}_{\mathrm{m}\mathcal{RLA}} \big(\mathfrak{g}(E),\mathfrak{g}'_2\big) \arrow[r, "\nu_{E,\mathfrak{g}_2'}"] &
\mathrm{Hom}_{\mathcal{LY}}(E,\mathfrak{m}_2') \end{tikzcd}.\tag{34}\] Indeed, for all morphisms \(\vartheta_2:\mathfrak{g}(E_2)\to \mathfrak{g}'\) of RLA pairs we have \[\begin{align} \nu_{E_1,\mathfrak{g}'} \big(\vartheta_2\circ (\mathbf{F}\zeta)\big) & = & \big(\vartheta_2\circ (\mathbf{F}\zeta)\big)|_{E_1} =\vartheta_2|_{E_2}\circ \zeta
=\big(\nu_{E_2,\mathfrak{g}'}(\vartheta_2)\big) \circ \zeta,
\end{align}\] which shows that the left diagram in (34 ) commutes. Next, for every morphism \(\vartheta_1:\mathfrak{g}(E)\to \mathfrak{g}'_1\) of RLA pairs we get \[\begin{align} (\mathbf{G}\phi')\circ (\nu_{E,\mathfrak{g}'_1}(\vartheta_1)) & =& (\mathbf{G}\phi')\circ \left(\vartheta_1|_{E}\right) = \big(\phi'\circ \vartheta_1\big)|_{E} =
\nu_{E,\mathfrak{g}'_2}(\phi'\circ\vartheta_1),
\end{align}\] which shows that the right diagram in (34 ) commutes.
In order to find an inverse of \(\nu_{E,\mathfrak{g}'}\), we associate to each morphism \(\chi:E\to \mathfrak{m}'\) of Lie-Yamaguti algebras the morphism of RLA pairs \(\check{\chi}:\mathfrak{g}(E)\to \mathfrak{g}'\) defined in lemma 4 induced by the map in (30 ). We compute for all morphisms of Lie-Yamaguti algebras \(\chi:E\to \mathfrak{m}'\), for all morphisms of RLA-pairs \(\vartheta:\mathfrak{g}(E)\to
\mathfrak{g}'\), and for all \(v,v_1,v_2\in E\): \[\big( \nu_{E,\mathfrak{m}'}(\check{\chi})\big)(v) =\check{\chi}|_E(v) =\chi(v)\] and \[\begin{align} \big( \nu_{E,\mathfrak{m}'}(\vartheta)\big)^\vee(v) &=& \big(\nu_{E,\mathfrak{m}'}(\vartheta)\big)(v)= \vartheta|_E(v) =\vartheta(v),\\ \big( \nu_{E,\mathfrak{m}'}(\vartheta)\big)^\vee \big(v_1\wedge
v_2~\mathrm{mod}~\tilde{I}(E)\big) & = & \Big[\nu_{E,\mathfrak{m}'}(\vartheta)(v_1), \nu_{E,\mathfrak{m}'}(\vartheta)(v_2) \Big]'_{\mathfrak{h}'} =\Big[\vartheta|_E(v_1), \vartheta|_E(v_2) \Big]'_{\mathfrak{h}'}\\ & =
& \big[\vartheta(v_1),\vartheta(v_2) \big]'_{\mathfrak{h}'} =\vartheta\big([v_1,v_2]_{\mathfrak{h}(E)} \big) = \vartheta\big(v_1\wedge v_2~\mathrm{mod}~\tilde{I}(E)\big),
\end{align}\] which shows that the inverse of \(\nu_{E,\mathfrak{m}'}\) is the map \(\chi\mapsto \check{\chi}\), whence \(\nu\) is a natural
isomorphism, and thus \(\mathbf{F}\) is a left adjoint functor of \(\mathbf{G}\)..
The unit of the adjunction \(\boldsymbol{\eta}:I
\Rightarrow \mathbf{GF}\) is a natural transformation whose component \[\boldsymbol{\eta}_E:E\to \mathbf{GF}E=\mathbf{G}\mathfrak{g}(E)=E\] is given by \(\boldsymbol{\eta}_E
=\nu_{E,\mathfrak{g}(E)}(\mathrm{id}_{\mathfrak{g}(E)}) =\mathrm{id}_{\mathfrak{g}(E)}|_E=\mathrm{id}_E\) which is an isomorphism.
The counit of the adjunction \(\boldsymbol{\epsilon}:\mathbf{FG}
\Rightarrow I\) is a natural transformation whose component \[\boldsymbol{\epsilon}_\mathfrak{g}:
\mathfrak{g}(\mathfrak{m})\to \mathfrak{g}\] is given by \(\boldsymbol{\epsilon}_\mathfrak{g} ={\nu_{\mathfrak{m},\mathfrak{g}}}^{-1} (\mathrm{id}_\mathfrak{m}) =\mathrm{id}^\vee_\mathfrak{m}\). Hence, we have that
\[\label{EqCompCounitFirstAdjunction} \forall~z,z_1,z_2\in\mathfrak{m}:~~~ \boldsymbol{\epsilon}_\mathfrak{g}(z)=z~~~\mathrm{and}~~~
\boldsymbol{\epsilon}_\mathfrak{g}\big(z_1\wedge z_2~\mathrm{mod}~\tilde{I}_\mathfrak{m}\big) =[z_1,z_2]_\mathfrak{h}.\tag{35}\] Obviously, \(\mathrm{Im}(\boldsymbol{\epsilon}_\mathfrak{g})=\mathfrak{m}
\oplus K\mathrm{Span}\{[z_1,z_2]_\mathfrak{h}~|~z_1,z_2\in\mathfrak{m}\}
=\mathfrak{i}(\mathfrak{m})=\mathfrak{g}\). ◻
Remark: The construction shows that for a finite-dimensional LY-algebra \((E,T,R)\) its free reductive Lie algebra pair \(\mathbf{F}(E,T,R)= (\mathfrak{g}(E),\mathfrak{h}(E),E)\) is also finite-dimensional since \(\Lambda^2 E\) is obviously finite-dimensional.
In order to locate the enveloping Lie algebra \(\hat{\mathfrak{g}}(E)\) of a LY algebra \((E,T,R)\) in a categorical manner, we specialize to the subcategories \(\mathcal{LY}\mathrm{s}\), \(\mathrm{m}\mathcal{RLP}\mathrm{s}\), and \(\mathcal{RLP}\mathrm{s}\) of \(\mathcal{LY}\), \(\mathrm{m}\mathcal{RLP}\), and \(\mathcal{RLP}\), respectively, by restricting all morphism sets to those of all surjective morphisms. We get the following
Theorem 2. The assignment \(\hat{\mathfrak{g}}:\mathcal{LY}\mathrm{s}\to \mathrm{m}\mathcal{RLP}\mathrm{s}\) defined by the classical enveloping Lie algebra is a covariant functor which is right
adjoint to the obvious restriction \(\mathbf{G}_s:\mathrm{m}\mathcal{RLP}\mathrm{s}\to \mathcal{LY}\mathrm{s}\) of the functor \(\mathbf{G}\). The counit of this second adjunction
is an isomorphism, and the unit \(\boldsymbol{\eta}_{s\mathfrak{g}}\) is surjective.
As a consequence, we have the following compositioin of canonical surjective morphisms in \(\mathrm{m}\mathcal{RLP}\) for every \(\big(\mathfrak{g},\mathfrak{h}, \mathfrak{m}\big)\) in \(\mathrm{m}\mathcal{RLP}\) \[\label{EqSurjCanMorph} \begin{tikzcd} \mathfrak{g}(\mathfrak{m}) \arrow[r, twoheadrightarrow,
"\boldsymbol{\epsilon}_\mathfrak{g}"] & \mathfrak{g} \arrow[r, twoheadrightarrow, "\boldsymbol{\eta}_{s\mathfrak{g}}"] &\hat{\mathfrak{g}}(\mathfrak{m}) \end{tikzcd}\tag{36}\] (where \(\boldsymbol{\epsilon}_\mathfrak{g}\) is given by (35 ) and whose kernels both are central ideals. In the finite-dimensional case it follows that any of the three algebras in (36 ) is solvable (nilpotent) iff \(\hat{\mathfrak{g}}(\mathfrak{m})\) is solvable (nilpotent).
Proof. For any surjective morphism \(\psi:(E,T,R)\to (E',T',R')\) of Lie-Yamaguti algebras and for all \(v,v_1,v_2\in E\) we should like to define the action of the would-be functor \(\hat{\mathfrak{g}}\) on \(\psi\) by \(\big(\hat{\mathfrak{g}}\psi\big)(v):=v\) and, in view of Definition (10 ) of the subalgebra \(\hat{\mathfrak{h}}(E)\), by \[\label{EqDefFunctorGHatOnMorphHHatHHat} \big(\hat{\mathfrak{g}}\psi\big)\left(\hat{R}(v_1,v_2)\right):= \hat{R}'\big(\psi(v_1),\psi(v_2)\big).\tag{37}\] In order to show that this is well-defined, we use the fact that the subalgebra \(\hat{\mathfrak{h}}(E)\cong \Lambda^2E/\mathrm{Ker}(\hat{R})\) and show that the \(K\)-linear map \(\Lambda^2\psi:\Lambda^2E\to\Lambda^2E'\) maps the kernel of \(\hat{R}\) to the kernel of \(\hat{R}'\) so that \(\Lambda^2\psi\) passes to the quotient: indeed, let \(\alpha=\sum_{i=1}^Nv_i\wedge w_i\) be an arbitrary element of \(\mathrm{Ker}\big(\hat{R}\big)\), where \(v_1,\ldots,v_N,w_1,\ldots,w_N\in E\). We compute for all \(v\in E\) \[\begin{align} \Big(\hat{R}'\big(\big(\Lambda^2\psi\big)(\alpha)\big)\Big)\big(\psi(v)\big) & = & \sum_{i=1}^NR'\big(\psi(v_i),\psi(w_i)\big)\big(\psi(v)\big) = \sum_{i=1}^N\psi\big(R(v_i,w_i)v\big) = \psi\left(\hat{R}(\alpha)(v)\right) = 0. \end{align}\] Since \(\psi\) is surjective we have that \(\hat{R}'\big(\big(\Lambda^2\psi\big)(\alpha)\big)=0\) and thereby proving that \(\big(\Lambda^2\psi\big)\big(\mathrm{Ker}\big(\hat{R}\big)\big)\subset \mathrm{Ker}\big(\hat{R}'\big)\) which makes formula (37 ) well-defined. Note that the linear map \(\Lambda^2\psi\) is surjective whence \(\hat{\mathfrak{g}}\psi\) is automatically surjective. It is a routine check that \(\hat{\mathfrak{g}}\psi\) is a morphism of RLA pairs using (37 ), (12 ), (?? ) and (8 ). Next, in order to prove the adjunction \[\nu_{\mathfrak{g},E'}: \mathrm{Hom}_{\mathcal{LY}\mathrm{s}} \left(\mathbf{G}_\mathrm{s}(\mathfrak{g},\mathfrak{h},\mathfrak{m}), (E',T',R')\right) \to \mathrm{Hom}_{\mathrm{m}\mathcal{RLP}\mathrm{s}} \left((\mathfrak{g},\mathfrak{h},\mathfrak{m}), \hat{\mathfrak{g}}(E',T',R')\right),\] we set \((\nu_{\mathfrak{g},E'}(\chi))(z):=\chi(z)\), and \[\forall~z_1,z_2\in \mathfrak{m}:~~~~ (\nu_{\mathfrak{g},E'}(\chi))\left([z_1,z_2]_\mathfrak{h}\right) := -\hat{R}'\big(\chi(z_1),\chi(z_2)\big)\] which is shown to be well-defined in a completely analogous way as for (37 ) by observing that the map \(\Lambda^2\mathfrak{m}\to \mathfrak{h}:z_1\wedge z_2\mapsto [z_1,z_2]_\mathfrak{h}\) is surjective since \(\mathfrak{g}\) is \(\mathfrak{m}\)-generated, and that its kernel is mapped by \(\Lambda^2\chi\) to the kernel of \(\hat{R}'\) thanks to the surjectivity of \(\chi\). By a longer, but straight-forward computation, it can be checked that each \(\nu_{\mathfrak{g},E'}(\chi)\) is a morphism of \(\mathrm{m}\mathcal{RLP}\mathrm{s}\), that its restriction to \(\mathfrak{m}\) is the inverse of \(\nu_{\mathfrak{g},E'}\), and that it defines a natural isomorphism. Finally, it is easily shown using (14 ), (15 ), and (16 ) that \(\mathrm{Ker}(\hat{R})\) contains the two-sided ideal \(\tilde{I}\) of \(\tilde{\mathfrak{g}}(E)\) and that the quotient \(\mathrm{Ker}(\hat{R})/ \tilde{I}\) is central in \(\mathfrak{g}(E)\) which completes the proof of the theorem. ◻
arXiv:2510.03148.