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

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, ... , 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 Uq(sl2) 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 Uq(sl2), thereby producing a large number of relations for the algebra generated by the intermediate Casimir elements.