Cohomology of Invariant Drinfeld Twists on Group Algebras

We show how to compute a certain group H l (G) 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 H l (Ok(G)) of the Hopf algebra Ok(G) 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 : b A × b A → k is a k-valued G-invariant non-degenerate alternating bilinear form on the dual b A. When the ground field k is not algebraically closed, we use algebraic group techniques to reduce the computation of H l (G) to a computation over the algebraic closure. As an application of our results, we compute H l (G) for a number of groups.