Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf

TL;DR

This paper establishes an explicit algebra isomorphism between Hochschild cohomology of certain Hopf-Galois extensions and H-invariant subalgebras, generalizing classical results and computing specific examples involving quantum planes.

Contribution

It provides a new explicit chain map inducing an algebra isomorphism in Hochschild cohomology for Hopf-Galois extensions, extending classical group action results.

Findings

Explicit chain map for Hochschild cohomology isomorphism

Generalization of classical group action results

Computed Hochschild cohomology for quantum plane and Kac--Paljutkin algebra

Abstract

In this paper, we give an explicit chain map, which induces the algebra isomorphism between the Hochschild cohomology ${\bf HH}^{\bullet}(B)$ and the $H$-invariant subalgebra ${\bf H}^{\bullet}(A, B)^{H}$ under two mild hypotheses, where $H$ is a finite dimensional semisimple Hopf algebra and $B$ is an $H$-Galois extension of $A$. In particular, the smash product $B=A\#H$ always satisfies the mild hypotheses. The isomorphism between ${\bf HH}^{\bullet}(A\#H)$ and ${\bf H}^{\bullet}(A, A\#H)^{H}$ generalizes the classical result of group actions. As an application, Hochschild cohomology and cup product of the smash product of the quantum $(-1)$-plane and Kac--Paljutkin Hopf algebra are computed.