Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf
Liyu Liu, Wei Ren, Shengqiang Wang
TL;DR
The paper develops an explicit chain-map approach to identify Hochschild cohomology of Hopf–Galois extensions with $H$-invariants of a relative cohomology, under two mild hypotheses, and shows this is compatible with cup products. In the semisimple-H case, Stefan’s spectral sequences are leveraged and recast via a double complex to yield ${\mathbf{HH}}^{\bullet}(B) \cong {\mathbf{H}}^{\bullet}(A,B)^{H}$. For Koszul algebras, the authors relate ${\mathbf{HH}}^{\bullet}(A\#H)$ to $H$-invariants of ${\mathbf{H}}^{\bullet}(A^{!}\otimes (A\#H))$ and provide an explicit differential graded isomorphism between $A^{!}\otimes (A\#H)$ and ${\mathrm{Hom}}_{A^{e}}(K(A),A\#H)$. As a concrete demonstration, they compute the Hochschild cohomology and cup-product structure for the smash product of the quantum plane with the Kac–Paljutkin Hopf algebra, supplying explicit bases and multiplication tables. This provides a practical toolkit for computing cup products in Hochschild cohomology for broad classes of Hopf–Galois extensions and their smash products.
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.