Data
- Title: Higher Rank Askey–Wilson Algebras as Skein Algebras
- Authors: Juliet Cooke and Abel Lacabanne
- arXiv link: https://arxiv.org/abs/2205.04414
- Accepted in: Ann. Inst. Fourier (Grenoble)
- DOI:
Abstract
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.
Citation