The Classification Of Leonard Triples Of Racah Type

Let K denote an algebraically closed field of characteristic zero. Let 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*, As 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.Let (A,A*,Aε) denote a Leonard triple on V. If some associated Leonard triple system has Racah type, then the Leonard triple (A, A*, Aε) is said to have Racah type.In this thesis, we mainly study the Leonard triples of Racah type, and classify them up to isomorphism. As an application, we construct all Leonard triples of Racah type by using the universal enveloping algebra U(sl2) and its representation theory.The thesis is composed of five chapters and organized as follows:In chapter1, 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.In chapter2, we introduce the notions of Leonard triple and Leonard triple system and give some related results.In chapter3, we first introduce the notions of Leonard systems and Leonard pairs of Racah type and give some related results. Then, we give the Askey-Wilson relations and the Z3-symmetric Askey-Wilson relations for a Leonard pair of Racah type.In chapter4, we first introduce the notions of Leonard triple systems and Leonard triples of Racah type. Then, we solve the classification of Leonard triples up to isomor-phism.In chapter5, we construct all Leonard triples of Racah type by using the universal enveloping algebra U(sl2) and its representation theory.