Gerstenhaber-Schack and Hochschild cohomologies of Hopf algebras

Abstract

We show that the Gerstenhaber-Schack cohomology of a Hopf algebra determines its Hochschild cohomology, and in particular its Gerstenhaber-Schack cohomological dimension bounds its Hochschild cohomological dimension, with equality of the dimensions when the Hopf algebra is cosemisimple of Kac type. Together with some general considerations on free Yetter-Drinfeld modules over adjoint Hopf subalgebras and the monoidal invariance of Gerstenhaber-Schack cohomology, this is used to show that both Gerstenhaber-Schack and Hochschild cohomological dimensions of the coordinate algebra of the quantum permutation group are 3.

arXiv comment21 pages.V3: title has changed, because the main theoretical results relating Hochschild and Gerstenhaber cohomologies are in fact valid for any Hopf algebra. However, because of a gap in part of Theorem 5.2 of the previous version, the computations in Section 6 now only cover the cosemisimple examples