Quantum duality maps, skein algebras and their ensemble compatibility
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912213481029632 |
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| author | Ishibashi, Tsukasa Karuo, Hiroaki |
| author_facet | Ishibashi, Tsukasa Karuo, Hiroaki |
| contents | We generalize the quantum duality map $\mathbb{I}_{\mathcal{A}}$ of Allegretti--Kim [AK17] for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map $\mathbb{I}_{\mathcal{X}}$ on the dual side based on the quantum bracelets basis [Thu14, MQ23]. Our construction factors through reduced stated skein algebras, based on the quantum trace maps [Lê18] together with an appropriate way of \emph{skein lifting} of integral $\mathcal{A}$-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_19074 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum duality maps, skein algebras and their ensemble compatibility Ishibashi, Tsukasa Karuo, Hiroaki Geometric Topology Quantum Algebra Representation Theory We generalize the quantum duality map $\mathbb{I}_{\mathcal{A}}$ of Allegretti--Kim [AK17] for punctured closed surfaces to general marked surfaces. When the marked surface has no interior marked points, we investigate its compatibility with the quantum duality map $\mathbb{I}_{\mathcal{X}}$ on the dual side based on the quantum bracelets basis [Thu14, MQ23]. Our construction factors through reduced stated skein algebras, based on the quantum trace maps [Lê18] together with an appropriate way of \emph{skein lifting} of integral $\mathcal{A}$-laminations. We also give skein theoretic proofs for some expected properties of Laurent expressions, and positivity of structure constants for marked disks and a marked annulus. |
| title | Quantum duality maps, skein algebras and their ensemble compatibility |
| topic | Geometric Topology Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2305.19074 |