Gerstenhaber-Schack and Hochschild cohomologies of Hopf algebras

TL;DR

This paper explores the relationship between Gerstenhaber-Schack and Hochschild cohomologies of Hopf algebras, establishing bounds and equalities for their cohomological dimensions, with applications to quantum permutation groups.

Contribution

It demonstrates that Gerstenhaber-Schack cohomology determines Hochschild cohomology for Hopf algebras and applies this to quantum permutation groups.

Findings

Gerstenhaber-Schack cohomology bounds Hochschild cohomological dimension.

Equality of cohomological dimensions in cosemisimple Kac type Hopf algebras.

Both cohomological dimensions of quantum permutation group are 3.

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.