### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 050, 11 pages      arXiv:1504.00558      https://doi.org/10.3842/SIGMA.2015.050
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

### Embeddings of the Racah Algebra into the Bannai-Ito Algebra

Vincent X. Genest, Luc Vinet and Alexei Zhedanov
Centre de Recherches Mathématiques, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, QC, Canada, H3C 3J7

Received April 02, 2015, in final form June 25, 2015; Published online June 30, 2015

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.

Key words: Bannai-Ito polynomials; Bannai-Ito algebra; Racah polynomials; Racah algebra.

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