Monomial web basis for the SL(N) skein algebra of the twice punctured sphere
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arXiv
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2024
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| author | Cremaschi, Tommaso Douglas, Daniel C. |
| author_facet | Cremaschi, Tommaso Douglas, Daniel C. |
| contents | We give a new proof of a slightly modified version of a result of Queffelec--Rose, by constructing a linear basis for the $\mathrm{SL}(n)$ skein algebra of the twice punctured sphere for any non-zero complex number $q$, excluding finitely many roots of unity of small order. In particular, the skein algebra is a commutative polynomial algebra in $n-1$ generators, where each generator is represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the surface. This includes the case $q=1$, where the skein algebra is identified with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace map, due originally to Bonahon--Wong in the case $n=2$. Two consequences of our method are that the quantum trace map and the so-called splitting map embed the polynomial algebra into the Fock--Goncharov quantum higher Teichmüller space and the Lê--Sikora stated skein algebra, respectively, of the annulus. We end by discussing the relationship with Fock--Goncharov duality. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_04178 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monomial web basis for the SL(N) skein algebra of the twice punctured sphere Cremaschi, Tommaso Douglas, Daniel C. Geometric Topology Combinatorics Quantum Algebra Representation Theory 2020: 57K31, 57M15, 14J33, 14T99 We give a new proof of a slightly modified version of a result of Queffelec--Rose, by constructing a linear basis for the $\mathrm{SL}(n)$ skein algebra of the twice punctured sphere for any non-zero complex number $q$, excluding finitely many roots of unity of small order. In particular, the skein algebra is a commutative polynomial algebra in $n-1$ generators, where each generator is represented by an explicit $\mathrm{SL}(n)$ web, without crossings, on the surface. This includes the case $q=1$, where the skein algebra is identified with the coordinate ring of the $\mathrm{SL}(n)$ character variety of the twice punctured sphere. The proof of both the spanning and linear independence properties of the basis depends on the so-called $\mathrm{SL}(n)$ quantum trace map, due originally to Bonahon--Wong in the case $n=2$. Two consequences of our method are that the quantum trace map and the so-called splitting map embed the polynomial algebra into the Fock--Goncharov quantum higher Teichmüller space and the Lê--Sikora stated skein algebra, respectively, of the annulus. We end by discussing the relationship with Fock--Goncharov duality. |
| title | Monomial web basis for the SL(N) skein algebra of the twice punctured sphere |
| topic | Geometric Topology Combinatorics Quantum Algebra Representation Theory 2020: 57K31, 57M15, 14J33, 14T99 |
| url | https://arxiv.org/abs/2407.04178 |