[Paper Review] Hopf algebra deformations of binary polyhedral groups

Julien Bichon, Sonia Natale|arXiv (Cornell University)|Jul 10, 2009

Algebraic structures and combinatorial models35 references4 citations

TL;DR

This paper classifies semisimple Hopf algebras with a self-dual faithful irreducible comodule of dimension 2, showing they arise as deformations of binary polyhedral groups. It establishes that such Hopf algebras are either commutative group algebras of binary polyhedral groups or nontrivial deformations, with their representation categories described via ${\mathbb{Z}}_2$-equivariantizations of pointed fusion categories.

ABSTRACT

We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.

Motivation & Objective

• To classify semisimple Hopf algebras over an algebraically closed field of characteristic zero that admit a self-dual faithful irreducible comodule of dimension 2.
• To show that such Hopf algebras are either commutative group algebras of binary polyhedral groups or nontrivial deformations of them.
• To describe the category of representations of these Hopf algebras using ${\mathbb{Z}}_2$-equivariantization of pointed fusion categories.
• To strengthen the Nichols-Richmond theorem in the finite-dimensional context by identifying the structure of the Hopf subalgebra generated by a 4-dimensional simple subcoalgebra.
• To provide an algebraic characterization of quotient Hopf algebras of $\mathcal{O}_{-1}[\mathrm{SL}_2(k)]$, analogous to compact quantum subgroups of $\mathrm{SU}_{-1}(2)$.

Proposed method

• Use the theory of abelian extensions and cocentral exact sequences to analyze Hopf algebras $H$ with a 4-dimensional simple subcoalgebra $C$ corresponding to a 2-dimensional irreducible comodule.
• Apply the Frobenius-Schur indicator $\nu(V) = \pm 1$ to distinguish between commutative and noncommutative cases, leading to classification into group algebras or deformations.
• Employ the structure of the Hopf subalgebra $B = k[C\mathcal{S}(C)] \simeq k^\Gamma$, where $\Gamma$ is a non-cyclic finite subgroup of $\mathrm{PSL}_2(k)$ of even order.
• Use group cohomology and the Kac exact sequence to classify extensions $k^\Gamma \to H \to k\mathbb{Z}_2$, identifying the role of $H^2(N, k^\times)$ and the connecting homomorphism $d_2$.
• Characterize the representation categories of the resulting Hopf algebras as ${\mathbb{Z}}_2$-equivariantizations of pointed fusion categories associated to $\Gamma$.
• Construct explicit deformations $\mathcal{A}[\widetilde{\Gamma}]$ and $\mathcal{B}[\widetilde{\Gamma}]$ of binary polyhedral groups $\widetilde{\Gamma}$, showing that only $\mathcal{B}[\widetilde{I}]$ exists for the icosahedral group.

Experimental results

Research questions

• RQ1 What is the structure of a finite-dimensional semisimple Hopf algebra that admits a self-dual faithful irreducible comodule of dimension 2?
• RQ2 How can such Hopf algebras be classified in terms of group algebras and their deformations?
• RQ3 What is the role of the Frobenius-Schur indicator in distinguishing between commutative and noncommutative Hopf algebras in this class?
• RQ4 How are the representation categories of these Hopf algebras related to equivariantizations of pointed fusion categories?
• RQ5 Which binary polyhedral groups admit nontrivial Hopf algebra deformations, and what are their cohomological invariants?

Key findings

• Semisimple Hopf algebras with a self-dual faithful irreducible comodule of dimension 2 are either commutative group algebras $k^{\widetilde{\Gamma}}$ of non-abelian binary polyhedral groups $\widetilde{\Gamma}$ (when $\nu(V) = -1$), or nontrivial deformations $\mathcal{A}[\widetilde{\Gamma}]$ or $\mathcal{B}[\widetilde{\Gamma}]$ (when $\nu(V) = 1$).
• The Hopf subalgebra $B = k[C\mathcal{S}(C)]$ is isomorphic to $k^\Gamma$, where $\Gamma$ is a non-cyclic finite subgroup of $\mathrm{PSL}_2(k)$ of even order, and $|G[\chi]|$ divides 4, with $G[\chi]$ the stabilizer of the character $\chi$.
• When $V$ is self-dual and $\nu(V) = 1$, the Hopf algebra $H$ fits into a cocentral exact sequence $k \to k^\Gamma \to H \to k\mathbb{Z}_m \to k$, with $\Gamma$ a polyhedral group of even order.
• The category of $H$-comodules is equivalent to the ${\mathbb{Z}}_2$-equivariantization of a pointed fusion category, as shown in Corollary 5.12.
• For the binary icosahedral group $\widetilde{I}$, only the deformation $\mathcal{B}[\widetilde{I}]$ exists, while for $D_n$, $\mathcal{A}[\widetilde{D}_n]$ and $\mathcal{B}[\widetilde{D}_n]$ may occur.
• The group $\operatorname{Opext}(k^\Gamma, k\mathbb{Z}_2)/K$ is isomorphic to $1$ or $\mathbb{Z}_2$, depending on the injectivity of the connecting homomorphism $d_2: H^2(\Gamma, k^\times) \to H^2(\mathbb{Z}_2, \widehat{\Gamma})$, with $K$ parametrizing twisting deformations of the split extension.