Center of stated $\mathrm{SL}(n)$-skein algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911174739623936 |
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| author | Karuo, Hiroaki Wang, Zhihao |
| author_facet | Karuo, Hiroaki Wang, Zhihao |
| contents | In the paper, we show some properties of (reduced) stated $\mathrm{SL}(n)$-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated $\mathrm{SL}(n)$-skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated $\mathrm{SL}(n)$-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12520 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Center of stated $\mathrm{SL}(n)$-skein algebras Karuo, Hiroaki Wang, Zhihao Quantum Algebra Geometric Topology Representation Theory In the paper, we show some properties of (reduced) stated $\mathrm{SL}(n)$-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated $\mathrm{SL}(n)$-skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated $\mathrm{SL}(n)$-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially. |
| title | Center of stated $\mathrm{SL}(n)$-skein algebras |
| topic | Quantum Algebra Geometric Topology Representation Theory |
| url | https://arxiv.org/abs/2408.12520 |