Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 077, 36 pages      arXiv:2303.17677      https://doi.org/10.3842/SIGMA.2023.077

The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms

Nicolas Crampé a, Luc Frappat b, Loïc Poulain d'Andecy c and Eric Ragoucy b
a)  Institut Denis-Poisson CNRS/UMR 7013 - Université de Tours - Université d'Orléans, Parc de Grandmont, 37200 Tours, France
b)  Laboratoire d'Annecy-le-Vieux de Physique Théorique LAPTh, Université Savoie Mont Blanc, CNRS, F-74000 Annecy, France
c)  Laboratoire de mathématiques de Reims UMR 9008, Université de Reims Champagne-Ardenne, Moulin de la Housse BP 1039, 51100 Reims, France

Received April 12, 2023, in final form October 10, 2023; Published online October 18, 2023

Abstract
We propose a definition by generators and relations of the rank $n-2$ Askey-Wilson algebra $\mathfrak{aw}(n)$ for any integer $n$, generalising the known presentation for the usual case $n=3$. The generators are indexed by connected subsets of $\{1,\dots,n\}$ and the simple and rather small set of defining relations is directly inspired from the known case of $n=3$. Our first main result is to prove the existence of automorphisms of $\mathfrak{aw}(n)$ satisfying the relations of the braid group on $n+1$ strands. We also show the existence of coproduct maps relating the algebras for different values of $n$. An immediate consequence of our approach is that the Askey-Wilson algebra defined here surjects onto the algebra generated by the intermediate Casimir elements in the $n$-fold tensor product of the quantum group ${\rm U}_q(\mathfrak{sl}_2)$ or, equivalently, onto the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. We also obtain a family of central elements of the Askey-Wilson algebras which are shown, as a direct by-product of our construction, to be sent to $0$ in the realisation in the $n$-fold tensor product of ${\rm U}_q(\mathfrak{sl}_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

Key words: Askey-Wilson algebra; braid group.

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