Higher Rank Askey–Wilson Algebras as Skein Algebras

Cooke, Juliet; Lacabanne, Abel

doi:10.5802/aif.3729

Higher Rank Askey–Wilson Algebras as Skein Algebras
[Les algèbres d’Askey–Wilson de rang supérieur en tant qu’algèbres d’écheveaux]

Cooke, Juliet ¹ ; Lacabanne, Abel ²

¹ School of Mathematical Sciences, University Park, Nottingham, NG7 2RD (United Kingdom)
² Laboratoire de Mathématiques Blaise Pascal (UMR 6620), Université Clermont Auvergne, Campus Universitaire des Cézeaux, 3 place Vasarely, 63178 Aubière Cedex (France)

Annales de l'Institut Fourier, Online first, 63 p.

Abstract (VO)
Résumé

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 $𝒰_{q} ({𝔰𝔩}_{2})$ or the reflection equation algebra. We then use the isomorphism of the Kauffman bracket skein algebra of the $(n + 1)$ -punctured sphere with the $𝒰_{q} ({𝔰𝔩}_{2})$ invariants of the Alekseev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the $𝒰_{q} ({𝔰𝔩}_{2})$ invariants of the Alekseev 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.

Dans cet article, nous donnons une interprétation topologique et un calcul diagrammatique pour l’algèbre d’Askey–Wilson de rang $n - 2$ en donnant un isomorphisme explicite avec l’algèbre d’écheveaux d’une sphère privée de $n + 1$ points. Pour faire ceci, on considère l’algèbre d’Askey–Wilson comme sous-algèbre d’un produit tensoriel tressé de $n$ copies soit du groupe quantique $𝒰_{q} ({𝔰𝔩}_{2})$ soit de l’algèbre de l’équation de réflexion. On utilise ensuite l’isomorphisme de l’algèbre d’écheveaux de la sphère privée de $n + 1$ points avec les invariants sous $𝒰_{q} ({𝔰𝔩}_{2})$ de l’algèbre d’Aleeksev. On trouve également la dimension graduée de ces invariants et appliquons ceci pour trouver une présentation de l’algèbre d’écheveaux de la sphère privée de cinq points, ainsi que de l’algèbre d’Askey–Wilson de rang $2$ .

Reçu le : 2022-12-21
Révisé le : 2023-10-12
Accepté le : 2023-12-21
Première publication : 2025-08-01

DOI : 10.5802/aif.3729

Classification : 17B37, 57M27, 57N10
Keywords: Askey–Wilson algebras, Skein algebras
Mots-clés : Algèbres d’Askey–Wilson, Algèbres d’écheveaux

Affiliations des auteurs :

Cooke, Juliet ¹ ; Lacabanne, Abel ²

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Cooke, Juliet; Lacabanne, Abel. Higher Rank Askey–Wilson Algebras as Skein Algebras. Annales de l'Institut Fourier, Online first, 63 p.

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