Higher Rank Askey-Wilson Algebras as Skein Algebras

TL;DR

This paper establishes a topological and diagrammatic framework for higher rank Askey-Wilson algebras by linking them to skein algebras of punctured spheres, revealing new algebraic presentations and invariants.

Contribution

It provides an explicit isomorphism between higher rank Askey-Wilson algebras and skein algebras, offering a new topological perspective and concrete algebraic presentations.

Findings

Isomorphism between Askey-Wilson and skein algebras of punctured spheres

Dimension formulas for invariants of the Aleeksev moduli algebra

Presentation of the rank 2 Askey-Wilson algebra

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.