A skew group ring of $\mathbb{Z}/2\mathbb{Z}$ over $U({\mathfrak{sl}}_2)$, Leonard triples and odd graphs

Abstract

We employ a skew group ring of $\mathbb{Z}/2\mathbb{Z}$ over $U({\mathfrak{sl}}_2)$ to construct modules over the universal Bannai--Ito algebra. In addition, we give the conditions under which the defining generators act as Leonard triples on the resulting modules. As a combinatorial realization, we establish an algebra homomorphism from the universal Bannai--Ito algebra onto the Terwilliger algebra of an odd graph. This homomorphism provides a unified description of Leonard triples on all irreducible modules over the Terwilliger algebra.

28 pages