2026/07/31 by Nicolas Crampé, Sarah Post, Luc Vinet
Physics and Astronomy · Mathematics · #math-ph #math.MP
arxiv created 2026/07/31 · arxiv updated 2026/08/03
We construct an explicit embedding of the rank two Racah algebra \mathfrakR2 into the rank two Jacobi algebra \mathfrakJ2. Working in the differential model of \mathfrakJ2 on the triangle, we recall that the subalgebra structure of \mathfrakJ2 is organized by a pentagon whose four upper edges carry rank one Jacobi algebras and whose base carries a rank one Racah algebra. Promoting the two multiplication operators to Racah type generators by tridiagonalization turns every edge of the pentagon into a rank one Racah algebra and realizes \mathfrakR2 inside \mathfrakJ2. We determine the common eigenfunctions of different commuting pairs explicitly as products of Gauss hypergeometric functions and Jacobi polynomials. Different overlap coefficients are computed involving univariate Wilson polynomials as well as, for one pair of bases, bivariate Racah polynomials of Tratnik type. We also recall how \mathfrakJ2 arises as a contraction of the rank two Racah algebra \mathfrakR2.