A quantum generalization of the schur multiplier

One hundred years ago Schur defined the so-called multiplier of a group $G$ in order to classify the projective representations of $G$. Around 1990 Drinfeld introduced what are now called Drinfeld twists in order to classify quantum groups. Invariant Drinfeld twists form a group which can be defined for any Hopf algebra. I'll point out that this group for some Hopf algebra coincides with the Schur multiplier of a finite group. I'll show how to compute this group of invariant Drinfeld twists for the group algebra of a finite group $G$. The answer involves a Galois cohomology group, the group of automorphisms of $G$ preserving conjugacy classes, and the normal abelian subgroups of $G$ of central type. I'll illustrate this with several examples of well-known finite groups. My talk is based on joint work with Pierre Guillot published in [1].