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

In this paper, we give an explicit chain map, which induces the algebra isomorphism between the Hochschild cohomology ${\mathbf{H}\mathbf{H}}^{\bullet }(B)$ and the $H$-invariant subalgebra ${\mathbf{H}}^{\bullet }(A,B{)}^H$ 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 $B=A\mathrm{\#}H$ always satisfies the mild hypotheses. The isomorphism between ${\mathbf{H}\mathbf{H}}^{\bullet }(A\mathrm{\#}H)$ and ${\mathbf{H}}^{\bullet }(A,A\mathrm{\#}H{)}^H$ 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.