The Higher-Rank Askey-Wilson Algebra and Its Braid Group Automorphisms - Crampé, Nicolas; Frappat, Luc; Poulain d'Andecy, Loïc; Ragoucy, Eric (Symmetry, Integrability and Geometry: Methods and Applications, 2023)
We propose a definition by generators and relations of the rank − 2 Askey-Wilson algebra () for any integer , generalising the known presentation for the usual case =3. The generators are indexed by connected subsets of {1, …, }, and the simple and rather small set of defining relations is directly inspired by the known case of = 3. Our first main result is to prove the existence of automorphisms of () satisfying the relations of the braid group on + 1 strands. We also show the existence of coproduct maps relating the algebras for different values of . 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 -fold tensor product of the quantum group Uq(₂) or, equivalently, onto the Kauffman bracket skein algebra of the ( + 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 -fold tensor product of Uq(₂), thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.