The Classification Of Leonard Triples That Have Bannai/Ito Type And Odd Diameters

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 triple on V we mean an ordered triple of linear transformations A, A*, Aε in End(V) such that for each B∈{A. A*,ε}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. The diameter of the Leonard triple (A, A*, Aε) is defined to be one less than the dimension of V.In this thesis we define a family of Leonard triples said to have Bannai/Ito type. We classify up to isomorphism the Leonard triples that have Bannai/Ito type and odd diameter. This thesis is divided into the following four parts:In Part1, we introduce some basic concepts concerning about Leonard pairs and Leonard triples.In Part2, we define a family of Leonard systems said to have Bannai/Ito type and discuss some related properties.In Part3, we define a family of Leonard pairs said to have Bannai/Ito type. For a given Leonard pair (A, A*) that has Bannai/Ito type and odd diameter, we show that there exists a unique Aε∈End(V) such that A, A*, Aε satisfy the Z3-symmetric Askey-Wilson relations. We also give necessary and sufficient conditions for the triple (A, A*, Aε) to be a Leonard triple of Bannai/Ito type.In Part4, we classify up to isomorphism the Leonard triples and Leonard triple systems that have Bannai/Ito type and odd diameter.