On the Higher-Rank Askey-Wilson Algebras

Abstract: In the paper, the algebra $\mathscr{A}(n)$, which is generated by an upper triangular generating matrix with triple relations, is introduced. It is shown that there exists an isomorphism between the algebra $\mathscr{A}(n)$ and the higher-rank Askey-Wilson algebra $\mathfrak{aw}(n)$ introduced by Cramp\'e et al. Furthermore, we establish a series of automorphisms of $\mathscr{A}(n)$, which satisfy braid group relations and coincide with those in $\mathfrak{aw}(n)$.