Cohomology of Invariant Drinfeld Twists on Group Algebras

We show how to compute a certain group ${\mathrm H}^2_{\ell}(G)$ 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 ${\mathrm H}^2_{\ell}({\mathcal O}_k(G))$ of the Hopf algebra $\mathcal O_k(G)$ 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 $b:\widehat A\times \widehat A\rightarrow k^{\times}$ is a k^×-valued G-invariant nondegenerate alternating bilinear form on the dual $\widehat A$. When the ground fieldk is not algebraically closed, we use algebraic group techniques to reduce the computation ofHℓ²(G) to a computation over the algebraic closure. As an application of our results, we compute Hℓ²(G) for a number of groups.