Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf (2502.01967v1)

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.