On the Higher-Rank Askey-Wilson Algebras

In the paper, the algebra $\mathcal{A}(n)$, 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 $\mathcal{A}(n)$ and the higher-rank Askey-Wilson algebra $\mathfrak{a}\mathfrak{w}(n)$ introduced by Crampé et al. Furthermore, we establish a series of automorphisms of $\mathcal{A}(n)$, which satisfy braid group relations and coincide with those in $\mathfrak{a}\mathfrak{w}(n)$.