Embeddings of the Racah Algebra into the Bannai-Ito Algebra

Symmetry Integrability and Geometry-Methods and Applications

article

Vincent X. Genest¹  Luc Vinet¹  Alexei Zhedanov¹ 

[1] Centre de Recherches Mathématiques, Université de Montréal, Succ. Centre-Ville

关键词: Bannai–Ito polynomials;    Bannai–Ito algebra;    Racah polynomials;    Racah algebra;   

DOI  :  10.3842/SIGMA.2015.050

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown