Hopf Galois extensions of Hopf algebroids
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913393316724736 |
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| author | Han, Xiao Schauenburg, Peter |
| author_facet | Han, Xiao Schauenburg, Peter |
| contents | We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-)regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that if $N\subseteq P$ is a left $\mathcal{L}$-Galois extension and $Γ$ is a 2-cocycle of $\mathcal{L}$, then for the twisted comodule algebra ${}_ΓP$, $N\subseteq{}_ΓP$ is a left Hopf Galois extension of the twisted Hopf algebroid $\mathcal{L}^Γ$. We study twisted Drinfeld doubles of Hopf algebroids as examples for the Drinfeld twist theory. Finally, we introduce cleft extension and $σ$-twisted crossed products of Hopf algebroids. Moreover, we show the equivalence of cleft extensions, $σ$-twisted crossed products, and Hopf Galois extensions with normal basis properties, which generalize the theory of cleft extensions of Hopf algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hopf Galois extensions of Hopf algebroids Han, Xiao Schauenburg, Peter Quantum Algebra We study Hopf Galois extensions of Hopf algebroids as a generalization of the theory for Hopf algebras. More precisely, we introduce (skew-)regular comodules and generalize the structure theorem for relative Hopf modules. Also, we show that if $N\subseteq P$ is a left $\mathcal{L}$-Galois extension and $Γ$ is a 2-cocycle of $\mathcal{L}$, then for the twisted comodule algebra ${}_ΓP$, $N\subseteq{}_ΓP$ is a left Hopf Galois extension of the twisted Hopf algebroid $\mathcal{L}^Γ$. We study twisted Drinfeld doubles of Hopf algebroids as examples for the Drinfeld twist theory. Finally, we introduce cleft extension and $σ$-twisted crossed products of Hopf algebroids. Moreover, we show the equivalence of cleft extensions, $σ$-twisted crossed products, and Hopf Galois extensions with normal basis properties, which generalize the theory of cleft extensions of Hopf algebras. |
| title | Hopf Galois extensions of Hopf algebroids |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2406.11058 |