Embeddings of the Racah Algebra into the Bannai-Ito Algebra

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.

Contribution to the special volume for Luc Vinet's 60th birthday

References in corpus (1)

• Bispectrality of the Complementary Bannai-Ito Polynomials

Cited by in corpus (5)

• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• The Heun-Racah and Heun-Bannai-Ito algebras
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions
• The universal additive DAHA of type $(C_1^{\vee },C_1)$ and Leonard triples