Second invariant cohomology of some finite-dimensional Hopf algebras
Debashish Goswami, Kiran Maity
TL;DR
The paper develops a categorical framework for computing second invariant cohomology of finite-dimensional Hopf algebras by translating invariant 2-cocycles into unitary monoidal autoequivalences of corepresentation categories. It applies this method to two cases: the group ring of $G = Z_8 ⋊ Aut(Z_8)$, where the invariant cohomology has a nontrivial order-2 class, and the eight-dimensional Kac–Paljutkin algebra, where the invariant cohomology is trivial. The approach relies on explicit analysis of fiber functors and fusion rules, connecting cohomology to monoidal autoequivalences of $Corep(\mathcal{Q})$. The results provide a concrete, computer-free classification of $H^{2}_{uinv}$ and $H^{2}_{inv}$ for these examples, contributing to the understanding of twists and fiber functors in compact quantum groups and their duals.
Abstract
We use categorical description of the invariant 2-cohomology group of Hopf algebra to compute such cohomology for two finite dimensional Hopf algebras: the group ring of $Z_8\rtimes Aut(Z_8)$ and Kac-Paljutkin algebra. For the first of these two examples, our categorical approach helps to settle the problem of computing this cohomology, which was left open in by Guillot and Kassel (\cite{dtwist}), where only some partial information about this cohomology was obtained.