The ribbon category of representations of a crossed module
October 02, 2025
Abstract
The theory of representations of a crossed module is a direct generalization of the theory of representations of groups. For a finite group \(G\), the Drinfeld quantum double of the group \(G\) is a Hopf algebra that represents a special case of crossed module of finite groups. Here we study how to extend the construction of the Drinfeld quantum double for any other kind of crossed module of finite groups. This leads to a Hopf algebra \(D(G,H)\) that presents similarities with a Drinfeld double. We then study simple subalgebras of \(D(G,H)\) and give two isomorphisms for the decomposition into a product of simple subalgebras. We then study the category \(\cmod{D(G,H)}_{\boldsymbol{Fd}}\) of finite dimensional modules over \(D(G,H)\), which turns out to be isomorphic to the category of finite dimensional representations of finite crossed modules of groups. These categories being monoidal, we also study links between direct sums of simple objects and tensor products of simple objects and give some results for a Clebsch-Gordan formula. We, in this context, present and develop the character theory for representations of crossed modules of finite groups, and detail the proofs. We then study the category itself, which leads to some ribbon invariants.
1 Introduction↩︎
The concept of crossed module is an algebraic structure that exhibits connections with a variety of other algebraic structures. Initially introduced in the 1940s by J.H.C. Whitehead in the context of groups, it originally was used as a natural tool to classify topological spaces up to homotopy via fundamental groups. The category of crossed modules over an algebraic structure is notably complex, and understanding it has led to various interpretations. The connections with the third cohomology group and central extensions motivate the study of crossed modules, as the relation to the third cohomology group, for example, is recurrent within crossed modules of different algebraic structures (see [1], [2]). Crossed modules of groups have also been shown to appear in the study of 2-categories, in a way similar to the way the group \(\Aut{A}\) appears in a 1-category for each of its objects \(A\) ([3]).
Our focus here is on the notion of crossed modules for finite groups. As with many algebraic structures, understanding a crossed module of finite groups can be achieved through its representation category. This was introduced by Bantay in [4] as an example of a premodular category. This approach notably includes a character theory that is a direct generalization of the character theory of group representations. This character theory has also been studied in [5]. Given that the representation of groups is itself a complex subject, this generalization faces similar challenges, along with new ones. The main theorem of this article is that the category of finite dimensional representations of a finite crossed module of groups is isomorphic to a category of finite dimensional modules over a Hopf algebra that we explicit. The latter category naturally carries a structure of fusion and Ribbon category as the Hopf algebra is particularly well-behaved. This new result opens up the search for ribbon invariants defined via finite dimensional representations of finite crossed modules of groups ([6], chapter I, theorem 2.5, [7] theorem 4.5. page 72). Ribbon invariants are especially interesting as it has been shown that they lead to invariants of 3-manifolds ( [7] chapter 8 for example), as 3-manifolds are obtained via surgery along ribbons. The fusion structure is also interesting in that fusion categories are restrictive enough that a classification has been given ([8]). We then give new formulas for simple objects of this category before studying tensor products of simple objects directly and via the character theory. We develop the character theory that was introduced by Bantay in [4], give full demonstrations of his affirmations and compute some character tables as well as give a result about the general structure of these character tables. We tie the theory given by Bantay to the previously-discussed Hopf algebra and our discussion about simple objects to give new simplified expressions for Clebsch-Gordan coefficients.
This work fully translates the foundations of finite dimensional representations of a given finite crossed module of groups into a theory of finite dimensional modules over an explicit Hopf algebra, with emphasis on the character theory and the notion of character table.
The next step in the study of finite dimensional representations of finite crossed modules of groups would be to determine the extent of its applications. It is a good example of premodular category, of fusion and ribbon category. Its applications in quantum topology has yet to be studied more extensively, especially in conformal field theory where it may have a place as the generalization of the Drinfeld quantum double. As the Drinfeld quantum double exists for more than just finite groups, it might also be interesting to see what is the right generalization and which of the results still hold when we try generalizing to other kinds of Drinfeld doubles. Moreover the notions of crossed modules are well defined in a semi-abelian context ([9]), the same approach might lead to results to some extents depending on which crossed modules we look at. A starting point might be to look at representation of crossed module of Lie algebras, which should naturally lead to representations of crossed module of finite dimensional Hopf algebras over \(\cC\), considering the Cartier-Gabriel classification ([10]).
Acknowledgments↩︎
I would first like to greatly thank Friedrich Wagemann for leading this research project, his availability and his guidance. I also am deeply thankful to Christian Blanchet for showing us that our category of interest was most likely a ribbon category, as well as its leads and contribution for ribbon invariants.
2 Preliminaries↩︎
A crossed module of finite groups, or finite crossed module of groups, is a quadruplet \(X=(G,H,\mu, \gamma)\) where \(G\), \(H\) are both finite groups, \(\mu\) is a left action of \(G\) over \(H\), written as \(\mu(g)(h)=g\cdot h\), via group automorphisms, and \(\gamma\) is a group morphism from \(H\) to \(G\) such that the following axioms are satisfied.
\(\gamma(g\cdot h)=g\gamma(h)g\mm\)
\(\gamma(h)\cdot n=hnh\mm\)
For example, for any finite group \(G\) there is a finite crossed module \((G,G,\conj{\_},\Id_G)\), where \(\conj{\_}\) is the action by conjugation. Another crossed module that always exists when we have a finite group \(G\) with neutral element \(1\) is \((G, \{1\}, g\mapsto \Id_{\{1\}}, 1\mapsto 1 )\). This trivial example will turn out to be quite important as his representations will exactly be the representations of the group \(G\), embedding the theory of finite group representations in our theory of finite crossed module of groups representations. More generally, a finite group \(G\) with normal subgroup \(H\) also gives rise to a finite crossed module of groups \((G,H,\conj{\_}, i)\), where \(i\) is the inclusion and \(\conj{\_}\) is the conjugation by elements of \(G\) on \(H\). Lastly, to give another kind of example, we also have, for any finite group \(G\), a finite crossed module of groups \((\Aut{G}, G, \Id_{\Aut{G}}, g\mapsto \conj{g} )\).
From a finite crossed module of groups \(X=(G, H, \mu, \gamma)\) arises a category of representations over a field \(\Kk\) that we will write \(\mM(X)\). The objects in this category are defined as follows.
A representation of \(X\) is a \(\Kk\)-vector space \(V=\dsum_{h\in H} V_h\) together with projectors \(P(h)\) such that \(V\) is a representation of the finite group \(G\) via a group morphism \(\map{Q}{G}{GL(V)}\) and \[P(g\cdot h)=Q(g)P(h)Q(g\mm)\] holds.
A morphism in this category is a map required to preserve both the \(H\)-grading and the \(G\)-action.
We can endow \(\mM(X)\) with the structure of tensor category via \((V,P,Q)\ot (W,P',Q')=(V\ot W,P'', Q'')\). Here, for \(n\in H\), \(P''(n)=\dsum_{h\in H}P(h)\ot P'(h\mm n)\) and \(Q''\) is simply the diagonal action of \(G\) over \(V\ot W\). This tensor category is also braided thanks to the \(\gamma\) morphism. The braiding is the map \[\begin{align} \mapp{c_{V,W}}{V\ot W}{W\ot V}{v\ot w}{\dsum_{n\in H}Q_W(\gamma(n))w\ot P_V(n)v.} \end{align}\]
We will actually focus in this work on \(\mM(X)_{\boldsymbol{Fd}}\), the full subcategory of finite dimensional representations (i.e. the underlying vector space is finite dimensional) of the crossed module \(X\).
3 A Hopf algebra that represents a crossed module of groups↩︎
3.1 The construction↩︎
In his book [11], Kassel gives a way to form new Hopf algebras from previous ones and actions that respect the structures. The Drinfeld quantum double of any Hopf algebra is such an example of Hopf algebra. A semi-direct product of group is a fitting context. This will be a generalization of the construction of the Drinfeld quantum double of a group. We will in this section use the theorems proved in [11] to construct an explicit Hopf algebra \(D(G,H)\) that will contain both \(\Kk G\) (the group algebra of \(G\) over \(\Kk\)) and \(\Kk[H]\) the dual of \(\Kk H\), for a crossed module of groups \((G,H\mu, \gamma)\) such that \(H\) is finite, as well as the relations between the two.
The construction is based on chapter IX of [11].
Let \(G\) be a group, \(H\) be a finite group such that \(G\) acts on \(H\) via automorphisms. The linearization of the action turns \(\Kk[H]\) into a left \(\Kk G\)-module-algebra by defining the module structure as over the bases \(g\cdot \kr{x}:=\kr{g\cdot x}\).
\((\Kk[H]\ot \Kk G, m, \eta, \Delta, \varepsilon, S)\) is a Hopf algebra, with the following mappings given on the bases
\[\mapp{m}{(\Kk[H]\ot \Kk G)\ot(\Kk[H]\ot \Kk G)}{\Kk[H]\ot \Kk G}{(\delta_x\ot a)\ot (\delta_y\ot b)}{\delta_{x,a\cdot y}\delta_x\ot (ab) }\]
\[\mapp{\eta}{\Kk}{\Kk[H]\ot \Kk G}{k}{k(1\ot 1)}\]
\[\mapp{\Delta}{\Kk[H]\ot \Kk G}{(\Kk[H]\ot \Kk G)\ot (\Kk[H]\ot \Kk G)}{\delta_x\ot a}{\dsum_{h\in H}(\delta_h\ot a)\ot(\delta_{h^{-1}x}\ot a)}\]
\[\mapp{\varepsilon}{\Kk[H]\ot \Kk G}{\Kk}{\delta_x\ot a}{\delta_{x,1}}\]
\[\mapp{S}{\Kk[H]\ot \Kk G}{\Kk[H]\ot \Kk G}{\delta_x\ot a}{\delta_{a^{-1}\cdot x\mm}\ot a^{-1}}\]
We can notice that the square of the antipode here is the identity morphism.
For \(\delta_x\in\Kk[H], a\in \Kk G,\) \(S^2(\delta_x\ot a)=S(\delta_{a^{-1}\cdot x\mm}\ot a^{-1})=\delta_{a\cdot(a^{-1}\cdot x\mm)\mm}\ot a=\delta_x\ot a\), so \(S^2=\Id\).
We will now write \(D(G,H):=\Kk[H]\ot\Kk G,\) for a semi-direct product of groups \(G\rtimes H,\) with \(H\) a finite group.
We can now give the main theorem here.
Theorem 1. Let \(X=(G,H,\mu,\gamma)\) be a finite crossed module of groups. Then \(\mM(X)\) and \(\cmod{D(G,H)}\) are isomorphic as monoidal categories.
This isomorphism restricts well to finite dimensional representations, meaning the categories \(\cmod{D(G,H)}_{\boldsymbol{Fd}}\) and \(\mM(X)_{\boldsymbol{Fd}}\) are also isomorphic.
One can easily see that taking \(H=\{1\}\subset G\) we find \(D(G,H)\cong \Kk G\) the algebra of the group \(G\). This remark justifies the inclusion of the theory of representations of finite groups into Bantay’s theory of representations of finite crossed modules of groups.
The category of representations of a crossed module of finite groups is known to be braided, therefore \(D(G,H)\) has an even richer structure, as stated in [11].
Let \((A,m,\eta,\Delta,\varepsilon)\) be a bialgebra. Then \(\cmod{A}\) is braided if and only if the bialgebra \(A\) is braided. Moreover, if \(c\) is the braiding in \(\cmod{A}\), then the \(R-\)matrix on \(A\) is given by \(R=\tau_{H,H}c_{H,H}(1\ot 1)\).
\(D(G,H)\) is braided, with \(R\)-matrix given by \(R=\dsum_{h\in H}\delta_h\ot 1\ot 1\ot\gamma(h).\)
The construction of \(D(G,H)\) does not require the full structure of crossed modules of groups, as the morphism we wrote \(\gamma\) does not appear in the construction (which reflects that it does not appear in objects in \(\mM(X)_{\boldsymbol{Fd}}\)). However, this morphism is used in the braiding, and therefore in the \(R\)-matrix.
3.2 Simple subalgebras and the algebra structure of D(G,H)↩︎
From now on, both groups \(G\) and \(H\) of the crossed module will always be finite, so that \(D(G,H)\) is of finite dimension \(\vert G\vert \vert H\vert .\)
3.2.1 Semisimplicity of \(D(G,H)\)↩︎
Theorem 2. If the characteristic of \(\Kk\) does not divide the order of \(G\), then \(D(G,H)\) is a semi-simple associative algebra.
Proof. This is a special case of a theorem from [12] that states that every Hopf algebra with \(S^2=\Id\) is semi-simple as an associative algebra.
The proof uses a projection onto an invariant complementary subspace.
If \(M\) is a \(D(G,H)\)-module and \(N\) is a sub-\(D(G,H)\)-module of \(M\), let \(p\) be a linear projection from \(M\) to the sub-vector space \(N\) of \(M.\) The projection \(P_N\) onto \(N\) from \(M\) as \(D(G,H)\)-modules is given by \[\begin{align} \mapp{P_N}{M}{N}{m}{\dfrac{1}{\vert G\vert }\dsum_{g\in G, h\in H}}(\kr{h}\ot g)\cdot p((\kr{g\mm\cdot h }\ot g\mm)\cdot m). \end{align}\] ◻
One can prove this theorem using an integral on the Hopf algebra. We can define this integral even in the case where the characteristic is non-zero, and we have a slightly better result about the characteristic of the field. The result from [12] would tell us that we need the characteristic of \(\Kk\) not to divide \(\vert G\vert \vert H\vert\), but considering the explicit construction of integrals from [13], this can be relaxed into not dividing \(\vert G\vert\).
Notice that if \(H\) is the trivial group, then we have the usual Maschke projection for group representations. We will focus on the structure of \(D(G,H)\) over \(\Cc\) so we do not have to worry about division algebras over our field.
3.2.2 The center of \(D(G,H)\) and formulae for simple subalgebras↩︎
Direct computations lead to an expression for a basis of the center of \(D(G,H)\).
\(\bB= \adaa{\dsum_{g \in G}\delta_{g\cdot z}\ot gcg^{-1}}_{ z\in H, c\in \Stab{G}{z}}\) is a basis of \(Z(D(G,H))\).
As an interesting consequence, we can give a proof for the cardinality of simple objects up to isomorphism in the category of finite dimensional representations of a crossed module of finite groups.
Let \(\hH\) be a set of representatives of \(H\backslash G\), the set of orbits of \(H\) under the action of \(G.\) \[\di{Z(D(G,H))}=\card{\{(x, [a]) \vert x\in \hH, [a] \text{ a conjugation class in }\Stab{G}{x}\}}\]
Proof. Let \(\aA=\{(x, [a]) \vert x\in \hH, [a] \text{ a class of conjugation in }\Stab{G}{x}\}.\) We are going to exhibit a bijection between \(\aA\) and \(\bB.\)
Let \[\mapp{\psi}{\aA}{\bB}{(x, [a])}{\dsum_{g\in G}\delta_{g\cdot x}\ot gag^{-1}}\].
First, we need to check that this is well defined. Let \(x\in \hH, a,b\in\Stab{G}{x}\) such that \([a]=[b].\) Then by definition there exists \(c\in \Stab{G}{x}\) such that \(a=cbc\mm\). Therefore \[\begin{align} \dsum_{g\in G}\delta_{g\cdot x}\ot gag\mm =\dsum_{g\in G}\delta_{g\cdot x}\ot gc b (gc )\mm=\dsum_{g\in G}\delta_{(gc\mm )\cdot x}\ot gbg\mm=\dsum_{g\in G}\delta_{g\cdot x}\ot gbg\mm \end{align}\] So \(\psi\) is well defined. Let us prove that it is surjective.
Let \(\beta=\dsum_{g\in G}\delta_{g\cdot x}\ot gag\mm\in \bB.\) Let \(\chi\) be the representative of \(G\cdot x\) that is in \(\hH.\) We can then write \(x=k\cdot \chi\), for some \(k\in G.\) We just have to check that \(k\mm ak\in \Stab{G}{\chi}\), which is true because \((k\mm ak)\cdot \chi=k\mm\cdot x=\chi.\) Follows
\[\begin{align} \psi(\chi,[k^{-1}ak])=\dsum_{g\in G}\delta_{g\cdot \chi}\ot gk\mm ak g\mm =\dsum_{g\in G}\delta_{(gk)\cdot \chi}\ot gag\mm=\beta. \end{align}\]
Lastly, the injectivity. Let \((x,[a]), (y,[b])\in \aA\) such that \(\psi(x,[a])=\psi(y,[b]).\) We rewrite as \[\begin{align} \dsum_{g\in G}\delta_{g\cdot x}\ot gag\mm=\dsum_{g\in G}\delta_{g\cdot y}\ot gbg\mm \end{align}\] Because the sums are written over a basis of \(D(G, H)\) as a vector space, a first consequence is that for this equality to be true, each term of the sum needs to be exactly equal to one term of the other sum. In particular, \(y\) needs to be in the orbit of \(x.\) Both being representatives, we have \(x=y.\) From this assertion, we can say that both \([a]\) and \([b]\) are conjugation classes in \(\Stab{G}{x}.\) Rewriting the equality, we have \[\begin{align} \dsum_{g\in G}\delta_{g\cdot x}\ot g(a-b)g\mm=0\ri \dsum_{g\in \Stab{G}{x}}gag\mm-gbg\mm=0\ri [a]=[b].\label{this32gives32us32information} \end{align}\tag{1}\] ◻
Equation (1 ) gives us information about the number of simple objects in \(\cfd{\cmod{D(G,H)}}\) the category of finite-dimensional left \(D(G,H)\)-modules.
Theorem 3. Let \(X=(G,H,\mu,\gamma)\) be a crossed module of finite groups, and \(\hH\) be a set of representatives for \(H\backslash G\). There is a one-to-one correspondence \[\adaa{\text{simple object in \mM(X)_{\boldsymbol{Fd}}}}\overset{1-1}{\longleftrightarrow}\adaa{(h, \chi_h)\vert \begin{matrix} h\in \hH, \chi_h\text{ is an irreducible character of the action of }\\ \Stab{G}{h} \text{ on } H\text{ induced by } \mu \end{matrix}}\]
We also have the following instance of Artin-Wedderburn theorem [14], p. 49, theorem 3.5.
Theorem 4. Let \(\hH\) be a system of representatives of \(H\backslash G\), and as before, for \(h\in \hH,\) \[\Cc\Stab{G}{h}=\dprod_{i=1}^{l_h}I_i^{\oplus n_i(h)}(h)\] be the decomposition of the group algebra \(\Cc\Stab{G}{h}\) into simple subalgebras. We have the following \(\Cc\)-algebra isomorphism. \[\begin{align} D(G,H)\cong \dprod_{h\in \hH}\dprod_{i=1}^{l_h}\text{Mat}_{\vert G\cdot h\vert n_i(h)}(\Cc). \end{align}\] The notation \(\text{Mat}_n(\Cc)\) refers to \(n\times n\) matrices with coefficients in \(\Cc\).
To be able to have an explicit form of the simple subalgebras of \(D(G,H)\) as such, we can also give the family of primitive central sum-1 orthogonal idempotents of \(D(G,H)\). An isomorphism then comes from [14], proposition a. p. 94. In the next subsection, we will use this result to deduce a Clebsch-Gordan formula (Proposition [Cleb-Gor]) in our context. We obtain it again via character theory in section 4.3.
Let \(\hH\) be a family of representatives for \(G\backslash H\). For each \(z\in \hH,\) let \(\{e_i^z\}_{1\leq i\leq n_z}\) be the family of central sum-1 primitive orthogonal idempotents for the group algebra \(\Cc\Stab{G}{z}\). Then the family \[\fF=\adaa{\dfrac{1}{\vert \Stab{G}{z}\vert }\dsum_{g\in G}\kr{g\cdot z}\ot g e_i^zg\mm}_{z\in \hH, 1\leq i\leq n_z}\] is a family of central primitive orthogonal sum-1 idempotents for the algebra \(D(G,H).\)
3.2.3 A Clebsch-Gordan formula for representations of a crossed module.↩︎
We now know what the simple representations of a given crossed module of finite groups \(X=(G,H,\mu, \gamma)\) are. We saw in the preliminaries that there exists a notion of tensor product in the category \(\mM(X)_{\boldsymbol{Fd}}\) of finite dimensional representations of \(X.\) We want to understand the decomposition into simple objects of the tensor product of two simple objects.
As we saw earlier, we can work in the category \(\cfd{\cmod{D(G,H)}}\) instead of \(\mM(X)_{\boldsymbol{Fd}}\). In the first category simple modules are known and to see if a simple finite dimensional \(D(G,H)\)-module appears in the decomposition of some arbitrary finite dimensional \(D(G,H)\) module \(M\), we just have to see if the corresponding idempotent central element is "orthogonal" to \(M\).
Direct computations lead to the following proposition.
Let \(I^a_i\) be the simple subalgebra corresponding to the idempotent element \(f_i^a\) for \(a\in \hH\) and \(i\in \llbracket 1, n_a\rrbracket.\) Let \(s,z,t\in \hH.\) Let \((i,j,k)\in \llbracket1, n_z\rrbracket\times\llbracket1,n_s\rrbracket\times\llbracket1, n_t\rrbracket\). Let \(\hH_{z,s}^t=\{(h_t,l_t)\in G^2 \vert (h_t\cdot z)(l_t\cdot s)=t\}.\) Then \(I_k^t\) is a sub-left \(D(G,H)\)-module of \(I_j^z\ot I_i^s\) if and only if both of the following are true.
\[t\in (G\cdot z)(G\cdot s).\]
\[\{h_te_i^zh_t\mm\ot l_te_j^sl_t\mm\}_{(h_t, l_t)\in\hH_{z,s}^t}\Delta(e_k^t)\not=\{0\}.\]
Proof. We work with equivalences. \(I^t_k\) is a sub-\(D(G,H)\)-module of \(I^z_j\ot I^s_i\) if and only if \[(\dsum_{g\in G}\kr{g\cdot t}\ot ge_k^tg\mm)\cdot (\dsum_{h,l\in G}\kr{h\cdot z}\ot he_i^zh\mm\ot \kr{l\cdot s}\ot le_j^sl\mm)\not=0.\]
Computing this, we have, writing \(e_k^t=\dsum_{a\in \Stab{G}{t}}\lambda_aa\), \[\begin{align} &(\dsum_{g\in G}\kr{g\cdot t}\ot ge_k^tg\mm)\cdot (\dsum_{h,l\in G}\kr{h\cdot z}\ot he_i^zh\mm\ot \kr{l\cdot s}\ot le_j^sl\mm) \\&=\dsum_{g,h,l\in G, m\in H}\dsum_{a\in \Stab{G}{t}}\lambda_a(\kr{m}\ot gag\mm)(\kr{h\cdot z}\ot he_i^zh\mm)\ot (\kr{m\mm g\cdot t}\ot gag\mm)(\kr{l\cdot s}\ot le_j^sl\mm) \\ &=\dsum_{g,h,l\in G, m\in H, a\in \Stab{G}{t}}\lambda_a\kr{m,(gag\mm h)\cdot z}\kr{m\mm g\cdot t, (gag\mm l)\cdot s}\kr{m}\ot gag\mm he_i^zh\mm\ot \kr{m\mm g\cdot t}\ot gag\mm le_j^sl\mm \\&=\dsum_{g,h,l\in G, a\in \Stab{G}{t}}\lambda_a\kr{(gag\mm h)\cdot z\mm g\cdot t, (gag\mm l)\cdot s}\kr{(gag\mm h)\cdot z}\ot gag\mm he_i^zh\mm\ot \kr{(gag\mm h)\cdot z\mm g\cdot t}\ot gag\mm le_j^sl\mm \\&=\dsum_{g,h,l\in G, a\in \Stab{G}{t}}\lambda_a\kr{(ag\mm h)\cdot z(ag\mm l)\cdot s,t}\kr{(gag\mm h)\cdot z }\ot gag\mm he_i^zh\mm \ot \kr{(gag\mm h)\cdot z\mm g\cdot t}\ot gag\mm le_j^sl\mm \end{align}\] From there we can see this is zero if \(\hH_{z,s}^t=\emptyset.\) Suppose it is not the case. If we do the change of variables \((ag\mm h, ag\mm l)=(h_t,l_t)\in \hH_{z,s}^t,\) we then have, resuming our computations \[\begin{align} &=\dsum_{g\in G, (h_t,l_t)\in \hH_{z,s}^t, a\in \Stab{G}{t}}\lambda_a\kr{(gh_t)\cdot z}\ot gh_te_i^zh_t\mm ag\mm\ot \kr{(gh_t)\cdot z\mm g\cdot t}\ot gl_te_j^sl_t\mm ag\mm\\ &=\dsum_{(h_t,l_t)\in \hH_{z,s}^t}\adap{\dsum_{g\in G}(\Delta(1\ot g))(\kr{(gh_t)\cdot z}\ot h_te_i^zh_t\mm\ot \kr{(gh_t)\cdot z\mm g\cdot t}\ot l_te_j^sl_t\mm)(\Delta(1\ot e_k^tg\mm))}\label{sum} \end{align}\tag{2}\]
Now, for a couple \((h_t, l_t)\in \hH_{z,s}^t\), \[\begin{align} &\kr{(gh_t)\cdot z}\ot \kr{(gh_t)\cdot z\mm g\cdot t}=\kr{(g'h_t)\cdot z}\ot \kr{(g'h_t)\cdot z\mm g'\cdot t}\\ &\ri\Delta(1\ot g)(h_te_i^zh_t\mm\ot l_te_j^sl_t\mm)\Delta(1\ot e^t_kg\mm)=\Delta(1\ot g')(h_te_i^zh_t\mm\ot l_te_j^sl_t\mm)\Delta(1\ot e^t_k{g'}\mm). \end{align}\] Indeed, suppose left-hand equality.
Then we have the system \[\left\{\begin{matrix} (gh_t)\cdot z=(g'h_t)\cdot z\\ (gh_t)\cdot z\mm g\cdot t=(g'h_t)\cdot z\mm g'\cdot t \end{matrix}\right. \ri \left\{\begin{matrix} (gh_t)\cdot z=(g'h_t)\cdot z\\ g\cdot t=g'\cdot t \end{matrix}\right. \ri g'g\mm\in \Stab{G}{t}\cap\Stab{G}{h_t\cdot z}.\] And as \((h_t\cdot z)(l_t\cdot s)=t\), we even have \(g' g\mm\in \Stab{G}{t}\cap\Stab{G}{h_t\cdot z}\cap \Stab{G}{l_t\cdot s}.\) In particular \(g'g\mm\) commutes with \(h_te_i^zh_t\mm, e_k^t\) and \(l_te_j^sl_t\mm.\)
Follows \[\begin{align} \Delta(1\ot g)(h_te_i^zh_t\mm\ot l_te_j^sl_t\mm)&\Delta(1\ot e^t_kg\mm)\\&=\Delta(1\ot g)(g\mm g'h_te_i^zh_t\mm{g'}\mm g\ot g\mm g'l_te_j^sl_t\mm{g'}\mm g)\Delta(1\ot g\mm g'e^t_k{g'}\mm gg\mm)\\ &=\Delta(1\ot g')(h_te_i^zh_t\mm\ot l_te_j^sl_t\mm )\Delta(1\ot e_k^t{g'}\mm). \end{align}\]
Similarly, for a given \(g\in G,\) \[\begin{align} &\kr{(gh_t)\cdot z}\ot \kr{(gh_t)\cdot z\mm g\cdot t}=\kr{(gh_t')\cdot z}\ot \kr{(gh_t')\cdot z\mm g'\cdot t}\\ &\ri\Delta(1\ot g)(h_te_i^zh_t\mm\ot l_te_j^sl_t\mm)\Delta(1\ot e^t_kg\mm)=\Delta(1\ot g)(h_t'e_i^z{h_t'}\mm\ot l_t'e_j^s{l_t'}\mm)\Delta(1\ot e^t_k{g}\mm). \end{align}\]
This is fairly easy to see, as as before here \(h_t'h_t\mm\in \Stab{G}{z}\), therefore commutes with \(e_i^z\), and \(l_t\) is determined up to \(\Stab{G}{s}\) entirely by \(h_t\) up to \(\Stab{G}{z}\) (i.e. changing \(l_t\) means changing it with \(l_t'\) such that \(l_tl_t'\in \Stab{G}{s}\)).
From there we deduce the sum in equation (2 ) has only one term per double graduation (first and third term in the tensor product), multiplied by some non-zero integer. In particular, the sum is zero if and only if each term is zero, that is, if \(\Delta(e_k^t)\) is right orthogonal to every element of the family \(\{h_te_i^zh_t\mm\ot l_te_j^sl_t\mm\}_{(h_t, l_t)\in\hH_{z,s}^t}.\) ◻
We lack an expression of the multiplicity of an occurrence of a simple module \(I_k^t\) in the tensor product \(I_j^z\ot I_i^s\) with this approach. For representations of finite groups, multiplicities are given by some scalar product of characters. The goal of this next section is to explore a similar concept for finite crossed modules of groups.
4 Characters on representations of crossed modules of finite groups↩︎
4.1 The structure of the vector space of characters of a crossed module↩︎
As for representations of finite groups, the representations of crossed modules of finite groups carry a character theory. The concept was first introduced by P. Bantay in [4]. He gives the following definition, which is a direct generalization of the theory of characters for finite group representations.
Let \(X=(G,H,\mu,\gamma)\) be a crossed module of finite groups, and \((V,P,Q)\) an object in \(\mM(X)_{\boldsymbol{Fd}}.\) The character of \((V,P,Q)\) is the \(\Cc\) valued function \[\require{physics} \mapp{\psi}{H\times G}{\Cc}{(m,g)}{\Tr_V(P(m)Q(g))}.\] This expression is invariant up to isomorphisms of representations of \(X\).
As for the irreducible characters of a group algebra, we will write \(\Irr(X)\) the set of characters of irreducible representations of \(X.\)
Bantay also defines what he calls class functions of a crossed module \(X\). This is a set of \(\Cc\)-valued functions that follow similar rules to those of characters on a crossed module; characters are class functions. Namely, the formal definition is the following.
Let \(X=(G,H,\mu,\gamma)\) be a crossed module of finite groups. A class function over \(X\) is a \(\Cc\)-valued function \(\psi\) such that
\(\psi(m,g)=0\) if \(g\cdot m\not=m\), for all \(m\in H, g\in G.\)
\(\psi(h\cdot m, h g h\mm)=\psi(m,g)\) for all \(m\in H, g,h\in G.\)
Now characters of a crossed module of groups are class functions as, for \(g,h\in G, m\in H, (V,P,Q)\in \mM(X)_{\boldsymbol{Fd}},\) using the cyclic property of the trace,
\[\require{physics} \begin{align} \Tr_V(P(m)Q(g))&=\Tr_V(P(m)^2Q(g))\\&=\Tr_V(P(m)Q(g)P(m))\\&=\Tr_V(P(m)P(g\cdot m)Q(g))\\&=\kr{m,g\cdot m}\Tr_V(P(m)Q(g)). \end{align}\]
\[\require{physics} \begin{align} \Tr_V(P(m)Q(g))&=\Tr_V(Q(h)P(m)Q(g)Q(h\mm))\\&=\Tr_V(P(h\cdot m)Q(hgh\mm). \end{align}\]
We will write \(\mathscr{C}l(X)\) the set of class functions of \(X\) over the field \(\Cc.\)
\(\mathscr{C}l(X)\) is a \(\Cc\)-vector space that is isomorphic to \(\Cc^{\vert G\vert \vert H\vert }\) as a vector space. It carries the natural scalar product \[\forall \psi_1, \psi_2\in \mathscr{C}l(X), \langle\psi_1,\psi_2\rangle=\dfrac{1}{\vert G\vert }\dsum_{m\in H, g\in G}\overline{\psi_1(m,g)}\psi_2(m,g).\] Here \(\overline{z}\) is the complex conjugate of the number \(z.\)
\(\mathscr{C}l(X)\) has an even richer structure. We define a multiplication that turns \(\mathscr{C}l(X)\) into a \(\Cc\)-algebra by the formula \[\forall \psi_1, \psi_2\in \mathscr{C}l(X), \forall m\in H, g\in G,\quad (\psi_1\psi_2)(m,g)=\dsum_{n\in H}\psi_1(n,g)\psi_2(n\mm m, g).\]
This multiplication makes sense in the context of characters, as \(\psi_{A\ot B}=\psi_A\psi_B.\)
The theory of characters for crossed modules of finite groups presents strong similarities with the theory of characters for finite groups, such as the properties of orthogonality for irreducible characters.
Using this within our construction, we have a formula for irreducible characters as, for a finite dimensional simple module \(I_i^s\) of \(D(G,H)\), the associated character is the mapping
\[\require{physics} \begin{align} \mapp{\chi_{i}^s}{G\times H}{\Cc}{(x,a)}{\Tr_{I_i^s}(m_{D(G,H)}((\kr{x}\ot a)\ot \_))} \label{irrcharmult} \end{align}\tag{3}\]
This formula explicitly confirms that irreducible characters are a basis of the vector space of all characters of the crossed module \(X\), as \(D(G,H)\) is semi-simple. Moreover, the dimension of this vector space is \(\di{Z(D(G,H))}\) the number of simple \(D(G,H)\)-modules up to isomorphism. This is also the dimension of \(\mathscr{C}l(X)\) and characters are class functions, so the vector space of characters is exactly the vector space of class functions. Notice that from there we can deduce that the multiplication is commutative for characters, as \(\mM(X)\) is braided and therefore for all \(A,B\) finite dimensional \(D(G,H)\)-modules, \(A\ot B\cong B\ot A.\)
From now on, we will write \(\psi_i^s\) for the irreducible character that corresponds to the irreducible character \(i\) of \(\Stab{G}{s}\). We will also write \(\psi_i^s(\kr{x}\ot a):=\psi_i^s(x,a)\), and the characters are then elements of \(D(G,H)^*\).
Let us write down the essential properties of characters, as shown in [4].
Let \(X=(G,H,\mu, \gamma)\) be a crossed module of finite groups and \(\Irr(X)\) the set of irreducible characters for its representation theory over \(\Cc\). We have the following properties.
\(\Irr(X)\) is an orthonormal basis of \(\mathscr{C}l(X)\) for the scalar product given above.
If \(A,B\) are finite dimensional representations of \(X\), with characters \(\chi_A\) and \(\chi_B\), then \(\chi_A+\chi_B\) is the character of \(A\oplus B.\)
Keeping the same hypotheses, \(\chi_A\chi_B\) is the character of \(A\ot B\).
The multiplicity of a finite dimensional representation \(A\) with character \(\chi_A\) in a finite dimensional representation \(B\) with character \(\chi_B\) is given by \(\langle \chi_A, \chi_B\rangle\).
If \(A\) is a finite dimensional representation of \(X\) with character \(\chi_A\), then its dimension is given by \(d=\dsum_{m\in H}\chi_A(m, 1)\).
Proof. We prove the first point.
We already saw that \(\Irr(X)\) was a basis of \(\mathscr{C}l(X).\) Let us check it is orthonormal. Now, for irreducible characters \(\psi_i^s\) and \(\psi_j^z\), they can only be non-zero over a single orbit in \(H\). For both to be simultaneously non-zero, it is necessary to have \(s=z\). Then, \(\psi_i^s(g\cdot s,ghg\mm)=\psi_i^s(s,h)\), so \[\begin{align} \langle \psi_i^s, \psi_j^s\rangle&=\frac{1}{\vert G\vert }\dsum_{g\in G, m\in H}\overline{\psi_i^s(m,g)}\psi_j^s(m,g)\\&=\frac{\vert G\cdot s\vert }{\vert G\vert }\dsum_{g\in \Stab{G}{s}}\overline{\psi_i^s(s,g)}\psi_j^s(s,g)\\&=\frac{1}{\vert \Stab{G}{s}\vert }\dsum_{g\in \Stab{G}{s}}\overline{\psi_i^s(s,g)}\psi_j^s(s,g)\\&=\kr{i,j}. \end{align}\] The last line comes from 3 .
The next two points come from the basic properties of the trace. The fourth point is a direct consequence of the first and second points, and lastly \[\require{physics} \begin{align} \di{A}&=\Tr_A(\Id_A)\\&=\dsum_{m\in H}\Tr_A(P(m)Q(1))\\&=\dsum_{m\in H}\chi_A(m,1). \end{align}\] ◻
These are all the prerequisites that we need to establish character tables of crossed modules of finite groups. We will now study their general form.
4.2 Character tables↩︎
Character tables of a crossed module of finite groups is a square table such that the entry \((i,j)\) represents the value of the \(i\)-th irreducible character evaluated over the \(j\)-th class of the crossed module \(X=(G,H, \mu, \gamma)\).
We can give a fairly precise description of the character table of a crossed module of finite groups. The tables are diagonal by block tables for which each diagonal block is the character table of the stabilizers of representants of the orbits. Indeed, we have already seen that an irreducible character is non-zero only over a single orbit of \(H\) under the action of \(G\), and then only over the stabilizer of elements of this particular orbit for the variable that lives in \(G\). Moreover, characters are class functions and, as such, their values are determined only by their values over a set of representatives of orbits.
Now we consider \(s\in H\). For \((V, P, Q)\) a finite dimensional irreducible representation of \(X\) with graduation in \(G\cdot s\), there is an induced \(\Stab{G}{s}\) representation over \(V\) given by \(\nu : g\mapsto (v\mapsto Q(g)P(s)v=P(s)Q(g)v)\). This is an irreducible representation of \(\Stab{G}{s}\) as if it were not the case, \((V,P,Q)\) would not be irreducible either for \(X\). Moreover, the character of the irreducible representation \(\nu\) is \[\require{physics} \Tr_{P(s)V}(Q(\_))=\Tr_V(Q(\_)P(s))=\Tr_V(P(s)Q(\_))=\psi_\nu^s(s,\_)\] as \(P(s)\) is a projection. Here \(\psi_\nu^s\) is the irreducible character of \(X\) that corresponds, i.e. \(V\cong I_i^s\) as \(D(G,H)\)-modules and \(V\cong e_i^s\Cc\Stab{G}{s}\) as \(\Cc\Stab{G}{s}\)-modules.
More visually, if \(\{s_k\}_{1\leq k\leq n}\) is a set of representatives for orbits of \(H\) under the action of \(G\),
| Orbits | \(s_1\) | \(s_2\) | \(\ldots\) | \(s_n\) | ||
|---|---|---|---|---|---|---|
| Conjugation classes of \(\Cc\Stab{G}{\_}\) | \([1]\) | \(\ldots\) | last conjugacy class | \(\ldots\) | ||
| \(\begin{matrix} \text{Characters of X} \\ \text{of the form } \psi_i^{s_1} \end{matrix}\) | \(\begin{matrix} \text{Character table of}\\ \text{the group } \Stab{G}{s_1} \end{matrix}\) | 0 | ||||
| \(\vdots\) | 0 | \(\ddots\) | 0 | |||
| \(\begin{matrix} \text{Characters of X} \\ \text{of the form } \psi_i^{s_n} \end{matrix}\) | 0 | \(\begin{matrix} \text{Character table of}\\ \text{the group } \Stab{G}{s_n} \end{matrix}\) | ||||
The character tables are displayed such that rows and columns correspond to their respective heads in the usual way, and the \(0\) represent tables of the appropriate size, such that each cell is filled with the number \(0\).
This is clearer with examples. We will give two of them to illustrate, one which is a Drinfeld quantum double and one that is not. Some more example have been computed in [5] and by myself while studying the subject.
Example 1 (The Drinfeld quantum double \(D(\Sym_3)\)). The conjugacy classes are well known. There are three of them : \([\Id], [(12)]\) and \([(123)]\). Their respective associated stabilizers, which here correspond to the centralizers of a representative, are \(\Sym_3\), \(\{\Id, (12)\}\cong \Zz/2\Zz\) and \(\{\Id, (123), (132)\}\cong \Zz/3\Zz\). Their respective table of characters are also well known, and therefore the table of characters of \(X=(\Sym_3, \Sym_3, \conj{\_},\Id_{\Sym_3})\) is the following. Here \(\omega\) is a primitive third root of unity in \(\Cc\). The simple representations of this Drinfeld double have been fully described in [15].
*|c||c|c|c!width 2ptc|c!width 2ptc|c|c| Orbit & & &
Conjugacy class & \([\Id]\) &\([(12)]\)&\([(123)]\)&\([\Id]\)&\([(12)]\)&\([\Id]\)&\([(123)]\)&\([(132)]\)
\(\psi_1^\Id\)& 1&1&1&0&0&0&0&0
\(\psi_2^\Id\)&1&-1&1&0&0&0&0&0
\(\psi_3^\Id\)&2&0&-1&0&0&0&0&0
\(\psi_1^{(12)}\)&0&0&0&1&1&0&0&0
\(\psi_2^{(12)}\)&0&0&0&1&-1&0&0&0
\(\psi_1^{(123)}\)&0&0&0&0&0&1&1&1
\(\psi_2^{(123)}\)&0&0&0&0&0&1&\(\omega\)&\(\omega^2\)
\(\psi_3^{(123)}\)&0&0&0&0&0&1&\(\omega^2\)&\(\omega\)*
Example 2. Here we look at \(X=(\Aut{D_4}\cong D_4, D_4, \text{ev}, x\mapsto x\_x\mm)\). This is not a Drinfeld quantum double even though \(\Aut{D_4}\cong D_4\). We only need to specify that \(\Stab{\Aut{D_4}}{s}\cong \Zz/2\Zz\), \(\Stab{\Aut{D_4}}{r}\cong \Zz/ 4\Zz\) and \(\Stab{\Aut{D_4}}{r^2}\cong\Stab{\Aut{D_4}}{1}\cong D_4\). The character table of \(X\) is therefore the following.
*|c||c|c|c|c|c!width 2ptc|c|c|c|c!width 2ptc|c|c|c!width 2ptc|c| Orbit & & &&
Conjugacy class & \([1]\)& \([r]\)& \([s]\)& \([r^2]\)& \([sr]\)&\([1]\)& \([r]\)& \([s]\)& \([r^2]\)& \([sr]\)&\([1]\)& \([r]\)& \([r^2]\)& \([r^3]\)& \([1]\)& \([s]\)
\(\psi_1^1\)& 1& 1& 1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&0
\(\psi_2^1\)& 1& -1& 1& 1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&0
\(\psi_3^1\)& 1& -1& -1& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&0
\(\psi_4^1\)& 1& 1& -1& 1& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&0
\(\psi_5^1\)& 2& 0& 0& -2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0&0
\(\psi_1^{r^2}\)& 0& 0& 0& 0& 0& 1& 1& 1& 1& 1& 0& 0& 0& 0& 0&0
\(\psi_2^{r^2}\)& 0& 0& 0& 0& 0& 1& -1& 1& 1& -1& 0& 0& 0& 0& 0&0
\(\psi_3^{r^2}\)& 0& 0& 0& 0& 0& 1& -1& -1& 1& 1& 0& 0& 0& 0& 0&0
\(\psi_4^{r^2}\)& 0& 0& 0& 0& 0& 1& 1& -1& 1& -1& 0& 0& 0& 0& 0&0
\(\psi_5^{r^2}\)& 0& 0& 0& 0& 0& 2& 0& 0& -2& 0& 0& 0& 0& 0& 0&0
\(\psi_1^r\)& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 1& 1& 0&0
\(\psi_2^r\)& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& i& -1& -i& 0&0
\(\psi_3^r\)& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -1& 1& -1& 0&0
\(\psi_4^r\)& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& -i& -1& i& 0&0
\(\psi_1^s\)& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1&1
\(\psi_2^s\)& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1&-1*
4.3 The Clebsch-Gordan formula via character theory↩︎
We will begin by reminding the following theorem, from Bantay in [4],
Theorem 5 (Bantay). The multiplicity of a finite dimensional simple \(D(G,H)\)-module \(I_k^t\) in the finite dimensional \(D(G,H)\)-module \(I_i^s\ot I_j^z\) is \[\begin{align} N_{(s,i),(z,j)}^{(t,k)}=\langle \psi_i^s\psi_j^z,\psi_k^t\rangle. \label{ClGo} \end{align}\qquad{(1)}\]
We can improve this formula with the previous considerations. We use the fact that an irreducible character is non-zero only over a single orbit.
\[\begin{align} N_{(s,i),(z,j)}^{(t,k)}&=\frac{1}{|G|}\dsum_{g\in G, n,m\in H}\psi_i^s(n,g)\psi_j^z(n\mm m,g)\overline{\psi_k^t(m,g)}\\ &=\frac{1}{\vert G\vert }\dsum_{g\in G,n,m\in H}\psi_i^s(n,g)\psi_j^z(m,g)\overline{\psi_k^t(nm,g)}\\ &=\dfrac{1}{\vert G\vert \vert \Stab{G}{s}\vert \vert \Stab{G}{z}\vert }\dsum_{a,b,g\in G}\psi_i^s(a\cdot s,g)\psi_j^z(b\cdot z,g)\overline{\psi_k^t((a\cdot s)(b\cdot z), g)}\\ &=\frac{1}{\vert G\vert \vert \Stab{G}{s}\vert \vert \Stab{G}{z}\vert }\dsum_{a,b,g\in G}\psi_i^s(s,a\mm ga)\psi_j^z(z,b\mm gb)\overline{\psi_k^t(s(a\mm b)\cdot z, a\mm ga)} \end{align}\] \[\begin{align} &=\frac{1}{\vert G\vert \vert \Stab{G}{s}\vert \vert \Stab{G}{z}\vert }\dsum_{a,b,g\in G}\psi_i^s(s,g)\psi_j^z(z,b\mm aga\mm b)\overline{\psi_k^t(s(a\mm b)\cdot z, g)}\\ &=\frac{1}{\vert \Stab{G}{s}\vert \vert \Stab{G}{z}\vert }\dsum_{a,g\in G}\psi_i^s(s,g)\psi_j^z(z,a\mm ga)\overline{\psi_k^t(s(a\cdot z),g)} \end{align}\]
From as soon as the second line, the condition we have already found about \((G\cdot s)(G\cdot z)\cap (G\cdot t)\not=\emptyset\) appears, as elements in the sum are non-zeros only if the variables \(n,m\) are elements of \(G\cdot s\), \(G\cdot z\) respectively and their product is in \(G\cdot t\).
These coefficients appear, as stated in [16] for the case of a Drinfeld double, as fusion coefficients for the Groethendieck algebra \(K_H(G)\) of \(G\mhyphen\)equivariant complex vector bundles over the finite group \(H\). This consideration ties our category to orbifold conformal field theories, \((K_H(G), \oplus, \ot )\) becoming the fusion algebra.
Consider \(I_i^s, I_j^z\) two complex finite dimensional irreducible representations of \(X=(G,H, \mu, \gamma )\) a crossed module of finite groups. Both are elements of \(K_H(G)\), and, paraphrasing relations above, \[\begin{align} I_i^s\ot I_j^z=\bigoplus_{t\in \hH}\bigoplus_{1\leq k \leq n_t} (I_k^t)^{\oplus N_{(s,i),(z,j)}^{(t,k)}} \end{align}\] with \(\hH=H/G\) orbits and for \(t\in \hH,\) \(n_t\) the number of conjugation classes of the group \(\Stab{G}{\overline{t}}\subset G\), which does not depend on the representative \(\overline{t}\) we choose for \(t\).
5 Quantum invariants↩︎
With all of our previous considerations, we see that our category \(\mM(X)_{\boldsymbol{Fd}}\) actually has a lot of structure. Indeed, as a category of finite-dimensional modules over a braided finite dimensional semi-simple Hopf algebra, \(\mM(X)_{\boldsymbol{Fd}}\) is a fusion category.
Moreover, the dual of an object \(M\), with basis \(\bB\), in \(\mM(X)_{\boldsymbol{Fd}}\), is the dual vector space \(M^*\) together with the left \(D(G,H)\)-module structure over a basis \[\mapp{\phi}{D(G,H)\ot M^*}{M^*}{\kr{h}\ot g\ot \kr{m}}{\dsum_{n \in \bB}\kr{m}((\kr{g\mm \cdot h\mm }\ot g\mm)\cdot n )\kr{n}.}\]
The evaluation and coevaluation morphisms are the usual ones.
Lastly, we give the following theorem. We refer to [11], Chapter XIV for definitions of ribbon element or algebra.
Theorem 6. \(D(G,H)\) is a ribbon algebra with ribbon element \[\theta=\dsum_{n\in H}\kr{n}\ot \gamma(n\mm).\]
Proof. The ribbon element is actually exactly the Drinfeld element of \(D(G,H)\) ([12], Definition 12.2.9) that is a central fixed point for the antipode \(S\). The fact that \(\theta\) is central comes from the property of the Drinfeld element ( [12], VIII.4 ) \[\begin{align} \forall x\in D(G,H), S^2(x)=\theta x \theta\mm \end{align}\] and \(S^2=\Id\). This is enough to prove that it is indeed a ribbon element of \(D(G,H)\). ◻
From \(\theta\) we can deduce all other possible ribbon element([12], Theorem 12.3.6.). Follows a corollary useful for generating less trivial invariants later.
Ribbon elements of \(D(G,H)\) are indexed by the subgroup \(C\) of \(G\) of central elements of order 2 that act trivially on \(H\). The set of all ribbon elements is \(R=\{(1 \ot c)\theta | c \in C\}\).
\(\mM(X)_{\boldsymbol{Fd}}\) is therefore a ribbon category and thus, using Rechetikhin-Turaev’s theorem ([6], Chapter I Theorem 2.5.), generates ribbon invariants for any \(c\in C\). This invariant is introduced as a covariant monoidal functor \(F\) from \(\textit{Rib}_{\mM(X)_{\boldsymbol{Fd}}}\) the category of \(\mM(X)_{\boldsymbol{Fd}}\)-colored ribbon tangles (as defined in [6], Chapter I. Subsection 2.2.) to \(\mM(X)_{\boldsymbol{Fd}}\). Over the elementary tangles, the functor has the following values. Let \((V, P_V, Q_V), (W, P_W, Q_W)\) be objects in \(\mM(X)_{\boldsymbol{Fd}}\). The blue strand is \(V\)-colored and the red one is \(W\)-colored. Morphisms are written in bases \(\bB_V\) (\(\bB_W)\) of the vector spaces \(V\) (\(W\)) (or the corresponding dual bases). Notice that \(F\) depends on \(c\in C\).
Figure 1: 
.
These values are enough to derive any value of \(F\) over any ribbon of \(\textit{Rib}_{\mM(X)_{\boldsymbol{Fd}}}\) ([6], Chapter I. Lemma 3.1.1. ). Since the category \(\mM(X)_{\boldsymbol{Fd}}\) is not modular, unless \(D(G,H)\) is the Drinfeld double of \(G\) [17], we cannot access stronger invariants such as those discussed in Chapter III of [6] without loss of generality.