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
• Published in: Ann. Inst. Fourier (Grenoble) 76 (2026), no. 3, 1055–1117
• 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

@article {MR5100593,
    AUTHOR = {Cooke, Juliet and Lacabanne, Abel},
     TITLE = {Higher {R}ank {A}skey--{W}ilson {A}lgebras as {S}kein
              {A}lgebras},
   JOURNAL = {Ann. Inst. Fourier (Grenoble)},
  FJOURNAL = {Universit\'e{} de Grenoble. Annales de l'Institut Fourier},
    VOLUME = {76},
      YEAR = {2026},
    NUMBER = {3},
     PAGES = {1055--1117},
      ISSN = {0373-0956,1777-5310},
   MRCLASS = {17B37 (57K31)},
  MRNUMBER = {5100593},
       DOI = {10.5802/aif.3729},
       URL = {https://doi-org.ezproxy.uca.fr/10.5802/aif.3729},
}