Higher Rank Askey-Wilson Algebras as Skein Algebras

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.

40 pages

References in corpus (3)

• The Universal Askey-Wilson Algebra
• The Relationship between Zhedanov's Algebra AW(3) and the Double Affine Hecke Algebra in the Rank One Case
• Relating stated skein algebras and internal skein algebras

Cited by in corpus (2)

• An Askey-Wilson Algebra of Rank 2
• Factorized $A_2$-Leonard pair