Higher Rank Askey-Wilson Algebras as Skein Algebras

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), Online first
• DOI: 10.5802/aif.3729

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

@unpublished{AIF_0__0_0_A17_0,
     author = {Cooke, Juliet and Lacabanne, Abel},
     title = {Higher {Rank} {Askey{\textendash}Wilson} {Algebras} as {Skein} {Algebras}},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2025},
     doi = {10.5802/aif.3729},
     language = {en},
     note = {Online first},
}