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

Let KK denote an algebraically closed field of characteristic zero. Let V   denote a vector space over KK with finite positive dimension. A Leonard pair on V   is an ordered pair of linear transformations in End(V)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 paper, we first 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. Then we determine all Leonard triples extended from given totally bipartite Leonard pairs of q-Racah type. Finally, we classify up to isomorphism the totally bipartite Leonard triples of q-Racah type.