Embeddings of the Racah Algebra into the Bannai-Ito Algebra

TL;DR

This paper explores how the Racah algebra can be embedded into the Bannai-Ito algebra through two realizations, revealing structural relationships and invariance properties in superintegrable systems.

Contribution

It introduces two realizations of the Racah algebra within the Bannai-Ito algebra, including a quadratic combination approach and a connection via Casimir operators.

Findings

Quadratic combinations generate a central extension of the Racah algebra.

The embedding holds independently of the realization.

Invariance algebra of superintegrable systems is related through these embeddings.

Abstract

Embeddings of the Racah algebra into the Bannai-Ito algebra are proposed in two realizations. First, quadratic combinations of the Bannai-Ito algebra generators in their standard realization on the space of polynomials are seen to generate a central extension of the Racah algebra. The result is also seen to hold independently of the realization. Second, the relationship between the realizations of the Bannai-Ito and Racah algebras by the intermediate Casimir operators of the $\mathfrak{osp}(1|2)$ and $\mathfrak{su}(1,1)$ Racah problems is established. Equivalently, this gives an embedding of the invariance algebra of the generic superintegrable system on the two-sphere into the invariance algebra of its extension with reflections, which are respectively isomorphic to the Racah and Bannai-Ito algebras.