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Pré-Publication, Document De Travail Année : 2023

The higher-rank Askey--Wilson algebra and its braid group automorphisms

Résumé

We propose a definition by generators and relations of the rank $n-2$ Askey--Wilson algebra $aw(n)$ 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 $aw(n)$ 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 $U_q(sl_2)$ 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 $U_q(sl_2)$, thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.

Dates et versions

hal-04178915 , version 1 (08-08-2023)

Identifiants

Citer

Nicolas Crampe, Luc Frappat, Loic Poulain d'Andecy, Eric Ragoucy. The higher-rank Askey--Wilson algebra and its braid group automorphisms. 2023. ⟨hal-04178915⟩
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