Cohomology of invariant Drinfeld twists on group algebras

TL;DR

This paper computes the group of invariant Drinfeld twists on finite group algebras, linking it to lazy cohomology and involving automorphisms and bilinear forms, with explicit examples provided.

Contribution

It provides a method to compute the group of invariant Drinfeld twists, connecting it to lazy cohomology and automorphism groups, with explicit classifications and examples.

Findings

The group of invariant Drinfeld twists is isomorphic to the second lazy cohomology group.

Explicit descriptions involve automorphisms and bilinear forms on abelian subgroups.

Several concrete examples illustrate the theoretical results.

Abstract

We show how to compute a certain group of equivalence classes of invariant Drinfeld twists on the algebra of a finite group G over a field k of characteristic zero. This group is naturally isomorphic to the second lazy cohomology group of the Hopf algebra of k-valued functions on G. When k is algebraically closed, the answer involves the group of outer automorphisms of G induced by conjugation in the group algebra as well as the set of all pairs (A, b), where A is an abelian normal subgroup of G and b is a k^*-valued G-invariant non-degenerate alternating bilinear form on the Pontryagin dual of A. We give a number of examples.