Johnson graphs as slices of a hypercube and an algebra homomorphism from the universal Racah algebra into $U(\mathfrak{sl}_2)$

TL;DR

This paper establishes a new algebra homomorphism from the universal Racah algebra to U(sl_2) using Johnson graphs as slices of a hypercube, revealing connections to Leonard triples and module decompositions.

Contribution

It introduces a novel algebra homomorphism from the Racah algebra to U(sl_2) based on Johnson graph structures and explores its implications for module theory and Leonard triples.

Findings

The homomorphism $lat$ maps the Racah algebra into $U(rak{sl}_2)$.

Finite-dimensional $U(rak{sl}_2)$-modules are completely reducible as $ e$-modules.

Leonard triples arise from the second dual distance operator of the hypercube and Johnson graph decompositions.

Abstract

From the viewpoint of Johnson graphs as slices of a hypercube, we derive a novel algebra homomorphism $\sharp$ from the universal Racah algebra $\Re$ into $U(\mathfrak{sl}_2)$. We use the Casimir elements of $\Re$ to describe the kernel of $\sharp$. By pulling back via $\sharp$ every $U(\mathfrak{sl}_2)$-module can be viewed as an $\Re$-module. We show that for any finite-dimensional $U(\mathfrak{sl}_2)$-module $V$, the $\Re$-module $V$ is completely reducible and three generators of $\Re$ act on every irreducible $\Re$-submodule of $V$ as a Leonard triple. In particular, Leonard triples can be constructed in terms of the second dual distance operator of the hypercube $H(D,2)$ and a decomposition of the second distance operator of $H(D,2)$ induced by Johnson graphs.