Noetherian and affine properties of quantum moduli and $\mathfrak{g}$-skein algebras
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866912567323000832 |
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| author | Baseilhac, Stéphane Faitg, Matthieu Roche, Philippe |
| author_facet | Baseilhac, Stéphane Faitg, Matthieu Roche, Philippe |
| contents | We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring. If the surface has punctures, we prove also that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group $U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skein algebra when $\mathfrak{g}=\mathfrak{sl}_2$. We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00396 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Noetherian and affine properties of quantum moduli and $\mathfrak{g}$-skein algebras Baseilhac, Stéphane Faitg, Matthieu Roche, Philippe Quantum Algebra 17B37, 20G42, 57R56 We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring. If the surface has punctures, we prove also that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group $U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skein algebra when $\mathfrak{g}=\mathfrak{sl}_2$. We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras. |
| title | Noetherian and affine properties of quantum moduli and $\mathfrak{g}$-skein algebras |
| topic | Quantum Algebra 17B37, 20G42, 57R56 |
| url | https://arxiv.org/abs/2302.00396 |