Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras
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2025
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| _version_ | 1866915937422147584 |
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| author | Huang, Yimin Mi, Zhongkai Qi, Tiancheng Wu, Quanshui |
| author_facet | Huang, Yimin Mi, Zhongkai Qi, Tiancheng Wu, Quanshui |
| contents | For any affine Hopf algebra $H$ which admits a large central Hopf subalgebra, $H$ can be endowed with a Cayley-Hamilton Hopf algebra structure in the sense of De Concini-Procesi-Reshetikhin-Rosso. The category of finite-dimensional modules over any fiber algebra of $H$ is proved to be an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ of $H$. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ such that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property, it is proved that if the zero locus of a discriminant ideal of $(H,C,\text{tr})$ is non-empty then it contains the orbit of the identity element of the affine algebraic group $\text{maxSpec}C$ under the left (or right) winding automorphism group action. Its proof relies on the fact that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property if and only if the $\overline{\varepsilon}$-Chevalley locus of $(H,C)$ coincides with $\text{maxSpec}C$.
Then, we provide a description of the zero locus of the lowest discriminant ideal of $(H,C,\text{tr})$. It is proved that the lowest discriminant ideal of $(H,C,\text{tr})$ is of level $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))+1$, where $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$ is the Grothendieck ring of the finite-dimensional Hopf algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ and $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))$ is the Frobenius-Perron dimension of $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$. Some recent results of Mi-Wu-Yakimov about lowest discriminant ideals are generalized. We also prove that all the discriminant ideals are trivial if $H$ has the Chevalley property. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_21879 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras Huang, Yimin Mi, Zhongkai Qi, Tiancheng Wu, Quanshui Quantum Algebra Category Theory Rings and Algebras Representation Theory 16G30, 16T05, 16D60, 17B37, 18D20, 18M05 For any affine Hopf algebra $H$ which admits a large central Hopf subalgebra, $H$ can be endowed with a Cayley-Hamilton Hopf algebra structure in the sense of De Concini-Procesi-Reshetikhin-Rosso. The category of finite-dimensional modules over any fiber algebra of $H$ is proved to be an indecomposable exact module category over the tensor category of finite-dimensional modules over the identity fiber algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ of $H$. For any affine Cayley-Hamilton Hopf algebra $(H,C,\text{tr})$ such that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property, it is proved that if the zero locus of a discriminant ideal of $(H,C,\text{tr})$ is non-empty then it contains the orbit of the identity element of the affine algebraic group $\text{maxSpec}C$ under the left (or right) winding automorphism group action. Its proof relies on the fact that $H/\mathfrak{m}_{\overline{\varepsilon}}H$ has the Chevalley property if and only if the $\overline{\varepsilon}$-Chevalley locus of $(H,C)$ coincides with $\text{maxSpec}C$. Then, we provide a description of the zero locus of the lowest discriminant ideal of $(H,C,\text{tr})$. It is proved that the lowest discriminant ideal of $(H,C,\text{tr})$ is of level $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))+1$, where $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$ is the Grothendieck ring of the finite-dimensional Hopf algebra $H/\mathfrak{m}_{\overline{\varepsilon}}H$ and $\text{FPdim}(\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H))$ is the Frobenius-Perron dimension of $\text{Gr}(H/\mathfrak{m}_{\overline{\varepsilon}}H)$. Some recent results of Mi-Wu-Yakimov about lowest discriminant ideals are generalized. We also prove that all the discriminant ideals are trivial if $H$ has the Chevalley property. |
| title | Chevalley property and discriminant ideals of Cayley-Hamilton Hopf Algebras |
| topic | Quantum Algebra Category Theory Rings and Algebras Representation Theory 16G30, 16T05, 16D60, 17B37, 18D20, 18M05 |
| url | https://arxiv.org/abs/2506.21879 |