The Racah algebra: An overview and recent results

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 $\mathfrak{s}\mathfrak{u}(1,1)$ in $\mathcal{U}(\mathfrak{s}\mathfrak{u}(1,1){)}^{\otimes n}$. 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.