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 $${\triangle_q}$$▵q of the Askey–Wilson algebras was recently studied. In this paper, we consider a family of infinite-dimensional $${\triangle_q}$$▵q-modules. By the universal property of these $${\triangle_q}$$▵q-modules, we classify the finite-dimensional irreducible $${\triangle_q}$$▵q-modules when q is not a root of unity.