Totally Bipartite Leonard Pairs And Totally Bipartite Leonard Triples Of Q-Racah Type

Let K denote an algebraically closed field of characteristic zero. Let V denote a vector space over K with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in End(V) such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. Whenever these tridiagonal matrices are bipartite, the Leonard pair is said to be totally bipartite. The notion of a Leonard triple and the corresponding notion of totally bipartite are similarly defined.In this thesis, we mainly discuss the classification of totally bipartite Leonard pairs, the extension of Leonard triples from given totally bipartite Leonard pairs of q-Racah type and the classification of totally bipartite Leonard triples of q-Racah type. This thesis is composed of five chapters and organized as follows:In Chapter1, we mainly introduce the notions of Leonard pairs, Leonard systems and Leonard triples, and give some related results.In Chapter2, we mainly introduce the notions of the anticommutator spin algebra A, the universal enveloping algebra U(sl2), the quantum algebra Uq(so3) and their irreducible representations, and give some related facts.In Chapter3, we classify up to isomorphism the totally bipartite Leonard pairs. The classification reveals that these Leonard pairs are of the q-Racah, Krawtchouk, or Bannai/Ito type.In Chapter4, we determine all Leonard triples extended from given totally bipartite Leonard pairs of q-Racah type.In Chapter5, we classify up to isomorphism the totally bipartite Leonard triples of q-Racah type.