Higher Rank Askey-Wilson Algebras as Skein Algebras
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866912600265064448 |
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| author | Cooke, Juliet Lacabanne, Abel |
| author_facet | Cooke, Juliet Lacabanne, Abel |
| contents | In this paper we give a topological interpretation and diagrammatic calculus for the rank $(n-2)$ Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of $n$ copies of either the quantum group $\mathcal{U}_q{(\mathfrak{sl}_2)}$ or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere with the $\mathcal{U}_q{(\mathfrak{sl}_2})$ invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the $\mathcal{U}_q{(\mathfrak{sl}_2})$ invariants of the Aleeksev moduli algebra and apply this to finding a presentation of the skein algebra of the five-punctured sphere and hence also find a presentation for the rank $2$ Askey-Wilson algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_04414 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Higher Rank Askey-Wilson Algebras as Skein Algebras Cooke, Juliet Lacabanne, Abel Quantum Algebra In this paper we give a topological interpretation and diagrammatic calculus for the rank $(n-2)$ Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of $n$ copies of either the quantum group $\mathcal{U}_q{(\mathfrak{sl}_2)}$ or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere with the $\mathcal{U}_q{(\mathfrak{sl}_2})$ invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the $\mathcal{U}_q{(\mathfrak{sl}_2})$ invariants of the Aleeksev moduli algebra and apply this to finding a presentation of the skein algebra of the five-punctured sphere and hence also find a presentation for the rank $2$ Askey-Wilson algebra. |
| title | Higher Rank Askey-Wilson Algebras as Skein Algebras |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2205.04414 |