On monoidal autoequivalences of the category of Yetter-Drinfeld modules over a group: The lazy case

TL;DR

This paper investigates the structure of monoidal autoequivalences of Yetter-Drinfeld modules over finite groups, providing a decomposition of automorphism groups and a Kuenneth-like formula for lazy cohomology, advancing understanding of their algebraic symmetries.

Contribution

It offers a decomposition of the Hopf algebra automorphism group of the Drinfeld double and proposes a Kuenneth-like formula for lazy cohomology, with partial proofs.

Findings

Decomposition of Hopf automorphism groups into three subgroups.

Bruhat decomposition for G=Z_p^n case.

Partial results on lazy cohomology and Brauer-Picard group calculation.

Abstract

An interesting open question is to determine the group of monoidal autoequivalences of the category of Yetter-Drinfeld modules over a finite group $G$, or equivalently the group of Bigalois objects over the dual of the Drinfeld double $DG$. In particular one would hope to decompose this group into terms related to monoidal autoequivalences for the group algebra, the dual group algebra and interaction terms. We report on our progress in this question: We first prove a decomposition of the group of Hopf algebra automorphisms of the Drinfeld double into three subgroups, which reduces in the case $G=\mathbb{Z}_p^n$ to a Bruhat decomposition of $\mathrm{GL}_{2n}(\mathbb{Z}_p)$. Secondly, we propose a Kuenneth-like formula for the Hopf algebra cohomology of $DG^*$ into three terms and prove partial results in the case of lazy cohomology. We use these results for the calculation of the Brauer-Picard group in the lazy case in [LP15].