On the Higher-Rank Askey-Wilson Algebras

TL;DR

This paper introduces a new algebra $\

Contribution

It establishes an isomorphism between the algebra $\\mathscr{A}(n)$ and the higher-rank Askey-Wilson algebra $\\mathfrak{aw}(n)$, and explores its automorphisms.

Findings

Proves the isomorphism between $\\mathscr{A}(n)$ and $\\mathfrak{aw}(n)$.

Identifies automorphisms satisfying braid group relations.

Connects algebraic structures with braid group symmetries.

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 $\mathfrak{aw}(n)$.