The Types And Constructions Of Leonard Triples With Quantum Parameter Being Not A Unit Root

Let K denote an algebraically closed field of characteristic zero and let V denote a vector space over K with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A, A*, Aεin End(V) such that for each B ∈ {A, A*, Aε} there exists a basis for V with respect to which the matrix representing B is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal.In this paper, we study Leonard triples with quantum parameter q, where q is not a root of unity. We show these Leonard triples are all of q-Racah type. Then we introduce the quantum algebra Uq(sl2) and its irreducible module, and construct Leonard triples with quantum parameter q, where q is not a root of unity.The thesis is composed of three chapters and organized as follows:In chapter 1, we first introduce the notions of Leonard pairs and Leonard systems and give some related results. Then, we give the parameter array of a Leonard system and the Askey?Wilson relations for a Leonard pair. Finally, we introduce the notions of Leonard triple and Leonard triple system and give some related results.In chapter 2, we first introduce the Leonard pairs having LB- T D type and give some related results. Then, we give the main results: the Leonard triples with quantum parameter being not a unit root only contain q-Racah typeIn chapter 3, we first introduce the Uq(sl2) and its irreducible module and give some related results. Then, we introduce the Leonard pairs on Uq(sl2)-module. Finally, we give the conditions some specific linear combination act on the irreducible Uq(sl2)-module as a Leonard triple, and the linear combinations are generated by the generators of Uq(sl2).