The Leonard Triples Having Classical Type

Let K denote an algebraically closed field of characteristic zero, V denote a nonzero vector space over K with finite dimension. By a Leonard triple on V we mean an ordered triple of linear transformations A,A*,Aε in 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 define a family of Leonard triples said to have classical type and show that these Leonard triples consist of two families:the Racah type and the Krawtchouk type. Moreover, we construct all Leonard triples that have Krawtchouk type from the universal enveloping algebra U(sl2).The paper is composed of three chapters and organized as follows:In Chapter 1, we introduce the notions of Leonard pairs and Leonard triples and give some related results.In Chapter 2, we first introduce the notions of Leonard pairs of classical type and give some related results. Then, we define a family of Leonard triples said to have classical type and show that these Leonard triples consist of two families:the Racah type and the Krawtchouk type.In Chapter 3, we first introduce the universal enveloping algebra U(sl2) and its rep-resentation theory. Then, we give the Z3-symmetric Askey-Wilson relations for a Leonard triple of Krawtchouk type. Finally, we construct all Leonard triples of classical type by using U(sl2).