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

TL;DR

This paper constructs modules over the universal Bannai--Ito algebra using a skew group ring over $U( ext{sl}_2)$, characterizes Leonard triples on these modules, and relates them to the Terwilliger algebra of odd graphs.

Contribution

It introduces a new module construction via skew group rings and establishes a homomorphism linking the Bannai--Ito algebra to the Terwilliger algebra of odd graphs.

Findings

Conditions for Leonard triples acting on modules are identified.

A homomorphism from the Bannai--Ito algebra to the Terwilliger algebra is constructed.

Unified description of Leonard triples on irreducible modules over the Terwilliger algebra is provided.

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.