Totally Bipartite/ABipartite Leonard pairs and Leonard triples of Bannai/Ito type

This paper is about three classes of objects: Leonard pairs, Leonard triples, and the finite-dimensional irreducible modules for an algebra $\mathcal{A}$. 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 the tridiagonal matrices are bipartite, the Leonard pair is said to be totally bipartite. A mild weakening yields a type of Leonard pair said to be totally almost bipartite. A Leonard pair is said to be totally B/AB whenever it is totally bipartite or totally almost bipartite. The notion of a Leonard triple and the corresponding notion of totally B/AB are similarly defined. There are families of Leonard pairs and Leonard triples said to have Bannai/Ito type. The Leonard pairs and Leonard triples of interest to us are totally B/AB and of Bannai/Ito type. Let $\mathcal{A}$ denote the unital associative $\K$-algebra defined by generators $x,y,z$ and relations\[xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.\]The algebra $\mathcal{A}$ has a presentation involving generators $x,y$ and relations\[x^{2}y+2xyx+yx^{2}=4y,\qquad y^{2}x+2yxy+xy^{2}=4x.\] In this paper we obtain the following results. We classify up to isomorphism the totally B/AB Leonard pairs of Bannai/Ito type. We classify up to isomorphism the totally B/AB Leonard triples of Bannai/Ito type. We classify up to isomorphism the finite-dimensional irreducible $\mathcal{A}$-modules. We show that these three classes of objects are essentially in one-to-one correspondence, and describe these correspondences in detail.