The universal DAHA of type and Leonard triples

Assume that (Formula presented.) is an algebraically closed field and q is a nonzero scalar in (Formula presented.) that is not a root of unity. The universal Askey–Wilson algebra (Formula presented.) is a unital associative (Formula presented.) -algebra generated by A, B, C and the relations state that each of (Formula presented.) is central in (Formula presented.) The universal DAHA (Formula presented.) of type (Formula presented.) is a unital associative (Formula presented.) -algebra generated by (Formula presented.) and the relations state that (Formula presented.) It was given an (Formula presented.) -algebra homomorphism (Formula presented.) that sends (Formula presented.) Therefore, any (Formula presented.) -module can be considered as a (Formula presented.) -module. Let V denote a finite-dimensional irreducible (Formula presented.) -module. In this paper, we show that A, B, C are diagonalizable on V if and only if A, B, C act as Leonard triples on all composition factors of the (Formula presented.) -module V.