The Constructions Of Leonard Pairs Associated With The Equitable Generators Of The Quantum Algebra U_q(f(K))

Let K denote an algebraically closed field of characteristic zero. Let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A, A*in End(V) such that for each B∈{A,A*} there exists a basis for V with respect to which the matrix representing B is diagonal and the matrix representing the other linear transformation is irreducible tridiagonal. Quantum algebra Ug(f(K)) is quantum deformation of sl2.Let x±1, y, z be the equitable generators of Uq(f(K)), and let x denote xm. Let V(d,a) be the irreducible Uq(f(K))-module of dimension d+1, where d is a nonnegative integer and a is the primitive2m-th root of unity.In this thesis we mainly discuss the relations between quantum algebra Uq(f(K)) and the theory of Leonard pairs, in particular, we construct the Leonard pairs by using the equitable generators of Uq(f(K)). This thesis is divided into the following four parts.In Part1, we introduce the basic concepts and properties concerning about a Leonard pair and a Leonard system.In Part2, we introduce some facts about the quantum algebra Uq(f(K)) and the irreducible Uq(f(K))-modu\es.In Part3, we define A, A*∈Uq(f(K)) as linear combinations for1,x,y,xy and1,y,z,yz, respectively. We give the necessary and sufficient conditions for the actions of A, A*on V(d, a) to be a Leonard pair. And we show that the Leonard pairs constructed above must be of q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk. dual q-Krawtchouk, quantum q-Krawtchouk or affine q-Krawtchouk.In Part4, for a given Leonard pair A, A*on V, we show that if A, A*is of one of q-Racah, q-Hahn, dual q-Hahn, q-Krawtchouk, dual q-Krawtchouk, quantum q-Krawtchouk and affine q-Krawtchouk types, then there exists an irreducible Uq(f(K))-m.odules struc-ture on V such that A (resp.A*) acts on V as a linear combination of1,x,y,xy (resp.1,y,z, yz).