The Racah algebra: An overview and recent results

Abstract

Recent results on the Racah algebra ${\mathcal{R}}_n$ of rank $n-2$ are reviewed. ${\mathcal{R}}_n$ is defined in terms of generators and relations and sits in the centralizer of the diagonal action of $\mathfrak{su}(1,1)$ in $\mathcal{U}(\mathfrak{su}(1,1){)}^{\otimes n}$. Its connections with multivariate Racah polynomials are discussed. It is shown to be the symmetry algebra of the generic superintegrable model on the $(n-1)$ - sphere and a number of interesting realizations are provided.

18 pages, survey paper based on talk at the conference Representation Theory XVI in Dubrovnik, 2019. Version 2: some typos corrected and references updated. Accepted for publication in Contemporary Mathematics