Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909293139197952 |
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| author | Ishibashi, Tsukasa Yuasa, Wataru |
| author_facet | Ishibashi, Tsukasa Yuasa, Wataru |
| contents | Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $Σ$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$-form $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q} \subset \mathscr{S}_{\mathfrak{sp}_4,Σ}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}_{\mathfrak{sp}_4,Σ}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundary-localization $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q}[\partial^{-1}]$ is included into a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sp}_4,Σ}$ that quantizes the function ring of the moduli space $\mathcal{A}_{Sp_4,Σ}^\times$. Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of Fomin--Pylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_01540 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$ Ishibashi, Tsukasa Yuasa, Wataru Geometric Topology Quantum Algebra Representation Theory 13F60, 57K31 (Primary), 57K20 (Secondary) Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$-case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{q}$ consisting of $\mathfrak{sp}_4$-webs on a marked surface $Σ$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$-form $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q} \subset \mathscr{S}_{\mathfrak{sp}_4,Σ}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}_{\mathfrak{sp}_4,Σ}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundary-localization $\mathscr{S}_{\mathfrak{sp}_4,Σ}^{\mathbb{Z}_q}[\partial^{-1}]$ is included into a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sp}_4,Σ}$ that quantizes the function ring of the moduli space $\mathcal{A}_{Sp_4,Σ}^\times$. Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of Fomin--Pylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$-webs, and give infinitely many supporting examples on a quadrilateral. |
| title | Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$ |
| topic | Geometric Topology Quantum Algebra Representation Theory 13F60, 57K31 (Primary), 57K20 (Secondary) |
| url | https://arxiv.org/abs/2207.01540 |