Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915135970344960 |
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| author | Liu, Liyu Ren, Wei Wang, Shengqiang |
| author_facet | Liu, Liyu Ren, Wei Wang, Shengqiang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_01967 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf Liu, Liyu Ren, Wei Wang, Shengqiang Rings and Algebras K-Theory and Homology Quantum Algebra 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. |
| title | Cup Products on Hochschild Cohomology of Hopf-Galois Extensions.pdf |
| topic | Rings and Algebras K-Theory and Homology Quantum Algebra |
| url | https://arxiv.org/abs/2502.01967 |