The rank two Jacobi algebra

2025/07/10 by Nicolas Crampé, Crampe, Nicolas, Satoshi Tsujimoto +5 · 2 citations
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Nonlinear Waves and Solitons #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2507.07766

openalex publication_date 2025/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The quadratic rank two Jacobi algebra is identified from the relations obeyed by the bispectral operators of the two variable Jacobi polynomials orthogonal on the triangle. It is seen to admit as subalgebras Racah and Jacobi algebras of rank one. The dual realizations in terms of differential operators in the variable representation and in terms of difference operators in the degree representation are provided. Structure relations for the two variable Jacobi polynomials are obtained as a by product.

Citations

• Griffiths polynomials of Racah type
• Factorized A₂-Leonard pair
• Realisations of Racah algebras using Jacobi operators and convolution identities
• A discrete realization of the higher rank Racah algebra
• Dunkl shift operators and Bannai-Ito polynomials
• Bivariate second--order linear partial differential equations and orthogonal polynomial solutions
• Hidden Symmetry of the Racah and Clebsch-Gordan Problems for the Quantum Algebra sl_q(2)

Cited by

• Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way
• Realization embeddings of the rank two Racah algebra into the rank two Jacobi algebra

Related

• Algebraic interpretation of the two-variable Jacobi polynomials on the triangle: the pentagonal way
• Jacobi polynomials and SU(2,2)
• Symmetric Jacobi Polynomials on a Triangle and Their Spectral Algebra
• The Jacobian algebras
• Realisations of Racah algebras using Jacobi operators and convolution identities