[Paper Review] On the irreducible representations of generalized quantum doubles

Sebastian Burciu|arXiv (Cornell University)|Feb 20, 2012

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper provides a complete classification of irreducible representations of generalized quantum doubles associated to semisimple Hopf algebras with a surjective skew pairing, generalizing the well-known parametrization of Drinfeld double representations of finite groups. It establishes a tensor product formula for these representations and shows that their Grothendieck ring exhibits a structure analogous to those arising from Green functors and cocentral extensions, extending known results on quantum doubles and Hochschild cohomology rings.

ABSTRACT

A description of all the irreducible representations of generalized quantum doubles associated to skew pairings of semisimple Hopf algebras is given. In particular a description of the irreducible representations of semisimple Drinfeld doubles is obtained. It is shown that the Grothendieck ring of these generalized quantum doubles have a structure similar to the rings that arise from Green functors. In order to do this we give a formula for the tensor product of any two such irreducible representations.

Motivation & Objective

• To classify all irreducible representations of generalized quantum doubles $D_{ u}(U,H)$ when $U$ and $H$ are semisimple Hopf algebras and the pairing induces a surjection $U \to H^{*\text{cop}}$.
• To generalize the parametrization of irreducible representations of Drinfeld doubles $D(G)$ for finite groups $G$ to the setting of generalized quantum doubles.
• To describe the tensor product structure of these irreducible representations in terms of induced modules over conjugate subalgebras.
• To show that the Grothendieck ring of $D_{ u}(U,H)$ has a ring structure analogous to those from Green functors and cocentral Hopf algebra extensions.
• To establish a categorical interpretation of the representation category as a relative center of a tensor functor induced by a Hopf algebra morphism.

Proposed method

• Apply Clifford theory for semisimple Hopf algebras to analyze representations of $D_{ u}(U,H)$ via the normal Hopf subalgebra $K(U)$, the largest central Hopf subalgebra of $U$.
• Use the universal grading group $G$ of $\mathrm{Rep}(U)$, defined by $K(U) = kG^*$, to parametrize irreducible representations via conjugacy classes and equivariant modules.
• Define the Hopf subalgebra $L(g) \subseteq H$ as a generalization of the centralizer algebra, and construct irreducible representations as $S_{g,M} = H \otimes_{L(g)} M$ for $M \in \mathcal{I}_g$.
• Derive a tensor product formula via double coset decomposition: $(H \otimes_{L(g)} M) \otimes (H \otimes_{L(h)} N) \cong \bigoplus_{x \in \mathcal{D}} H \otimes_{L({}^xgh)} P(x)$, where $P(x)$ is a tensor product of restricted and induced modules.
• Establish a ring surjection from the Grothendieck ring $K_0(D_{ u}(U,H))$ onto the centralizer $\mathrm{C}_{\mathbb{Z}G}(\mathbb{Z}F)$, showing structural similarity to rings from Green functors and cocentral extensions.
• Interpret the representation category $\mathrm{Rep}(D_{ u}(U,H))$ as the relative center of the image functor $f_*$ induced by a surjective Hopf algebra morphism $f: U \to H^{*\text{cop}}$.

Experimental results

Research questions

• RQ1 How can the irreducible representations of generalized quantum doubles $D_{ u}(U,H)$ be classified when $U$ and $H$ are semisimple and the pairing induces a surjection $U \to H^{*\text{cop}}$?
• RQ2 What is the structure of the tensor product of two irreducible representations of $D_{ u}(U,H)$, and how does it generalize the known formula for Drinfeld doubles of finite groups?
• RQ3 How does the Grothendieck ring of $D_{ u}(U,H)$ relate to known ring structures such as those from Green functors or cocentral extensions?
• RQ4 What is the categorical interpretation of $\mathrm{Rep}(D_{ u}(U,H))$ in terms of relative centers of tensor functors?
• RQ5 To what extent does the representation theory of $D_{ u}(U,H)$ mirror that of $D(G)$ for finite groups $G$?

Key findings

• All irreducible representations of $D_{ u}(U,H)$ are of the form $S_{g,M} = H \otimes_{L(g)} M$, where $g$ ranges over representatives of $F$-orbits in the universal grading group $G$ of $\mathrm{Rep}(U)$, and $M$ is an irreducible representation in $\mathcal{I}_g$.
• The tensor product of two irreducible representations is given by $ (H \otimes_{L(g)} M) \otimes (H \otimes_{L(h)} N) \cong \bigoplus_{x \in \mathcal{D}} H \otimes_{L({}^xgh)} P(x) $, where $\mathcal{D}$ is a set of double coset representatives and $P(x)$ is a module induced from the intersection of $L(g)$ and $L(h)$.
• The Grothendieck ring $K_0(D_{ u}(U,H))$ decomposes as $\bigoplus_{g \in \Gamma} K_0(\mathcal{B}_g)$, with multiplication satisfying $K_0(\mathcal{B}_g) K_0(\mathcal{B}_h) \subseteq \bigoplus_{x \in \mathcal{D}} K_0(\mathcal{B}_{{}^xgh})$, reflecting a structure similar to Green functors.
• There exists a surjective ring homomorphism $K_0(D_{ u}(U,H)) \to \mathrm{C}_{\mathbb{Z}G}(\mathbb{Z}F)$, sending $[S_{g,M}]$ to $(\dim M) \cdot s(g)$, where $s(g)$ is the orbit sum in $\mathbb{Z}G$, generalizing results from Cibils and Witherspoon.
• The representation category $\mathrm{Rep}(D_{ u}(U,H))$ is equivalent to the relative center of the tensor functor $f_*$ induced by a surjective Hopf algebra morphism $f: U \to H^{*\text{cop}}$, providing a categorical interpretation.
• The Grothendieck ring structure of $D_{ u}(U,H)$ matches the ring structure described in [30] for abelian cocentral extensions, confirming a deep structural analogy with known constructions in Hochschild cohomology and representation theory.