Suogang Gao

The Leonard triples extended from given Leonard pairs of Bannai/Ito type

Let denote an algebraically closed field of characteristic zero and let denote an even at least . Let and be by matrices. Then is a Leonard pair on of Bannai/Ito type. We determine all the matrices such that form a Leonard triple on .

The Leonard triples with quantum parameter being not a unit root

Let denote an algebraically closed field of characteristic zero and let denote a vector space over with finite positive dimension. By a Leonard triple on , we mean an ordered triple of linear transformations in such that for each there exists a basis for with respect to which the matrix representing is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal. In this paper, we consider Leonard triples with quantum parameter , where is not a root of unity. We show these Leonard triples are all of -Racah type.

Totally almost bipartite Leonard pairs and Leonard triples of q-Racah type

Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in 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 two tridiagonal matrices are almost bipartite, the Leonard pair is said to be totally almost bipartite. The notion of a Leonard triple and the corresponding notion of totally almost bipartite are similarly defined. Let q denote a quantum parameter of a Leonard pair and let ‘TAB’ be an abbreviation for ‘totally almost bipartite’. In this paper we show that a TAB Leonard pair with q equal to is of Bannai/Ito type, and a TAB Leonard pair with q being not a root of unity is of q-Racah type. Under the assumption that q is not a root of unity, we classify, up to isomorphism, the TAB Leonard pairs of q-Racah type and the TAB Leonard triples of q-Racah type.