Papers › Higher Rank Askey-Wilson Algebras as Skein Algebras
Higher Rank Askey-Wilson Algebras as Skein Algebras
Juliet Cooke, Abel Lacabanne
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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 𝒰_q(𝔰𝔩₂) or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the (n+1)-punctured sphere with the 𝒰_q(𝔰𝔩₂) invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the 𝒰_q(𝔰𝔩₂) 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.
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