This paper is about three classes of objects: Leonard triples,
distance-regular graphs and the modules for the anticommutator spin algebra.
Let \K denote an algebraically closed field of characteristic zero. Let $V$
denote a vector space over \K with finite positive dimension. A Leonard
triple on $V$ is an ordered triple of linear transformations in
$\mathrm{End}(V)$ 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 matrices representing the other two transformations are
irreducible tridiagonal. The Leonard triples of interest to us are said to be
totally B/AB and of Bannai/Ito type.
Totally B/AB Leonard triples of Bannai/Ito type arise in conjunction with the
anticommutator spin algebra $\mathcal{A}$, the unital associative \K-algebra
defined by generators $x,y,z$ and relations$xy+yx=2z,\qquad yz+zy=2x,\qquad
zx+xz=2y.$
Let $D\geq0$ denote an integer, let $Q_{D}$ denote the hypercube of diameter
$D$ and let $\tilde{Q}_{D}$ denote the antipodal quotient. Let $T$ (resp.
$\tilde{T}$) denote the Terwilliger algebra for $Q_{D}$ (resp.
$\tilde{Q}_{D}$).
We obtain the following. When $D$ is even (resp. odd), we show that there
exists a unique $\mathcal{A}$-module structure on $Q_{D}$ (resp.
$\tilde{Q}_{D}$) such that $x,y$ act as the adjacency and dual adjacency
matrices respectively. We classify the resulting irreducible
$\mathcal{A}$-modules up to isomorphism. We introduce weighted adjacency
matrices for $Q_{D}$, $\tilde{Q}_{D}$. When $D$ is even (resp. odd) we show
that actions of the adjacency, dual adjacency and weighted adjacency matrices
for $Q_{D}$ (resp. $\tilde{Q}_{D}$) on any irreducible $T$-module (resp.
$\tilde{T}$-module) form a totally bipartite (resp. almost bipartite) Leonard
triple of Bannai/Ito type and classify the Leonard triple up to isomorphism.Comment: arXiv admin note: text overlap with arXiv:0705.0518 by other author