Classifications And Constructions Of Totally Almost Bipartite Leonard Triples

The theory of Leonard pairs and Leonard triples etc is a new one to study association schemes,which is called the representation theory of Terwilliger algebras.This new theory has close connections with Lie algebras,quantum algebras and mathematical physics.In the representation theory of Terwilliger algebras,there are two families of Leonard pairs and two families of Leonard triples with strong combinatorial background.They are totally bipartite Leonard pairs,totally almost bipartite Leonard pairs and totally bipartite Leonard triples,totally almost bipartite Leonard triples,respectively.In this thesis,we classify the totally almost bipartite Leonard pairs and totally almost bipartite Leonard triples,respectively.For a given totally almost bipartite Leonard pair of Bannai/Ito type and a given totally almost bipartite Leonard pair of q-Racah type,we construct the corresponding Leonard triples,respectively.This thesis mainly includes the following results:1.We prove that there are only two families of totally almost bipartite Leonard pairs.They are of q-Racah type and Bannai/Ito type.Then under the assumption that q is not a root of unity,we classify the totally almost bipartite Leonard pairs of q-Racah type by using the representation theory of quantum enveloping algebra Uq?so₃?.2.Under the assumption that q is not a root of unity,we classify the totally almost bipartite Leonard triples of q-Racah type by using the representation theory of quantum enveloping algebra Uq?so₃?.3.Let K denote an algebraically closed field of characteristic zero.For a given totally almost bipartite Leonard pair?A,A*?of Bannai/Ito type on K^d+1,we determine all the matrices A^? ? Mat_d+1?K?such that?A,A*,A^??forms a Leonard triple on K^d+1.4.Let K denote an algebraically closed field of characteristic zero.For a given totally almost bipartite Leonard pair?A,A*?of q-Racah type on K^d+1,we determine all the matrices A^? ?Mat_d+1?K?such that?A,A*,A^??forms a Leonard triple on K^d+1.