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.

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

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Cited by in corpus (14)

• A higher rank Racah algebra and the ${\mathbb{Z}}_2^n$ Laplace-Dunkl operator
• Representations of the rank two Racah algebra and orthogonal multivariate polynomials
• Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero
• A superintegrable model with reflections on $S^{n-1}$ and the higher rank Bannai-Ito algebra
• Finite-dimensional irreducible modules of the Bannai--Ito algebra at characteristic zero
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• The Heun-Racah and Heun-Bannai-Ito algebras
• The dual pair $Pin(2n)\times \mathfrak{osp}(1|2)$, the Dirac equation and the Bannai-Ito algebra
• Coupling coefficients of $s{u}_q(1,1)$ and multivariate $q$-Racah polynomials
• The Racah Algebra as a Subalgebra of the Bannai-Ito Algebra
• Finite-dimensional modules of the universal Racah algebra and the universal additive DAHA of type $(C_1^{\vee },C_1)$
• A superintegrable model with reflections on $S^3$ and the rank two Bannai-Ito algebra
• The universal additive DAHA of type $(C_1^{\vee },C_1)$ and Leonard triples
• Matrix elements of $SO(3)$ in $s{l}_3$ representations as bispectral multivariate functions