Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910473325117440 |
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| author | Baseilhac, Stéphane Roche, Philippe |
| author_facet | Baseilhac, Stéphane Roche, Philippe |
| contents | We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^ε$ of the quantum graph algebra at a root of unity $ε$ of odd order, and show that $\mathcal{L}_{0,n}^ε$ and its invariant algebra under the quantum group $U_ε(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_04136 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1 Baseilhac, Stéphane Roche, Philippe Quantum Algebra 16R30, 17B37, 20G42, 57R56 We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^ε$ of the quantum graph algebra at a root of unity $ε$ of odd order, and show that $\mathcal{L}_{0,n}^ε$ and its invariant algebra under the quantum group $U_ε(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute. |
| title | Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1 |
| topic | Quantum Algebra 16R30, 17B37, 20G42, 57R56 |
| url | https://arxiv.org/abs/2106.04136 |