Finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

Let F denote an algebraically closed field and assume that q∈F is a primitive dth root of unity with d≠1,2,4. The universal Askey–Wilson algebra △_q is a unital associative F-algebra defined by generators and relations. The generators are A,B,C and the relations assert that each of [Formula presented] [Formula presented] commutes with A,B,C. We show that every finite-dimensional irreducible △_q-module is of dimension less than or equal to {difdis odd;d/2ifdis even. Moreover we provide an example to show that the bound is tight.