Smoothness of commutative Hopf algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912643427598336 |
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| author | Egami, Kensuke Masuoka, Akira Suzuki, Kenta |
| author_facet | Egami, Kensuke Masuoka, Akira Suzuki, Kenta |
| contents | Hopf algebras, most generally in a semisimple abelian symmetric monoidal category, are here supposed to be commutative but not to be of finite-type, and their (equivariant) smoothness are discussed. Given a Hopf algebra $H$ in a category such as above, it is proved that the following are equivalent: (i) $H$ is smooth as an algebra; (ii) $H$ is smooth as an $H$-comodule algebra; (iii) the product morphism $S_H^2(H^+) \to H^+$ defined on the 2nd symmetric power is monic. Working over a field $k$ of characteristic zero, we prove: (1) every ordinary Hopf algebra, i.e., such in the category $\mathsf{Vec}$ of vector spaces, satisfies the equivalent conditions (i)--(iii) and some others; (2) every Hopf algebra in the category $\mathsf{sVec}$ of super-vector spaces has a certain property that is stronger than (i). In the case where $\operatorname{char}k=p>0$, there are shown weaker properties of ordinary Hopf algebras and of Hopf algebras in $\mathsf{sVec}$ or in the ind-completion $\mathsf{Ver}_p^{\mathrm{ind}}$ of the Verlinde category. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19038 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smoothness of commutative Hopf algebras Egami, Kensuke Masuoka, Akira Suzuki, Kenta Rings and Algebras Algebraic Geometry 16T05, 14L15 Hopf algebras, most generally in a semisimple abelian symmetric monoidal category, are here supposed to be commutative but not to be of finite-type, and their (equivariant) smoothness are discussed. Given a Hopf algebra $H$ in a category such as above, it is proved that the following are equivalent: (i) $H$ is smooth as an algebra; (ii) $H$ is smooth as an $H$-comodule algebra; (iii) the product morphism $S_H^2(H^+) \to H^+$ defined on the 2nd symmetric power is monic. Working over a field $k$ of characteristic zero, we prove: (1) every ordinary Hopf algebra, i.e., such in the category $\mathsf{Vec}$ of vector spaces, satisfies the equivalent conditions (i)--(iii) and some others; (2) every Hopf algebra in the category $\mathsf{sVec}$ of super-vector spaces has a certain property that is stronger than (i). In the case where $\operatorname{char}k=p>0$, there are shown weaker properties of ordinary Hopf algebras and of Hopf algebras in $\mathsf{sVec}$ or in the ind-completion $\mathsf{Ver}_p^{\mathrm{ind}}$ of the Verlinde category. |
| title | Smoothness of commutative Hopf algebras |
| topic | Rings and Algebras Algebraic Geometry 16T05, 14L15 |
| url | https://arxiv.org/abs/2412.19038 |