Finite-dimensional irreducible modules of the universal Askey-Wilson algebra

Since the introduction of Askey-Wilson algebras by Zhedanov in 1991, the classification of the finite-dimensional irreducible modules of Askey-Wilson algebras remains open. A universal analog ${\mathrm{△}}_q$ of the Askey-Wilson algebras was recently studied. In this paper, we consider a family of infinite-dimensional ${\mathrm{△}}_q$-modules. By the universal property of these ${\mathrm{△}}_q$-modules, we classify the finite-dimensional irreducible ${\mathrm{△}}_q$-modules when $q$ is not a root of unity.