The generalized Racah algebra as a commutant

TL;DR

This paper characterizes the Racah algebra as a commutant in oscillator representations and links it to Howe duality, providing a framework for deriving superintegrable models on spheres.

Contribution

It establishes a new perspective on the Racah algebra as a commutant and connects it to Howe duality and superintegrable models.

Findings

R(n) is obtained as a commutant of o(2)^n in o(2n) oscillator representations.

R(n) relates to Casimir operators in recouplings of multiple su(1,1) copies.

Framework enables derivation of superintegrable models on spheres invariant under R(n).

Abstract

The Racah algebra $R(n)$ of rank $(n-2)$ is obtained as the commutant of the \mbox{$\mathfrak{o}(2)^{\oplus n}$} subalgebra of $\mathfrak{o}(2n)$ in oscillator representations of the universal algebra of $\mathfrak{o}(2n)$. This result is shown to be related in a Howe duality context to the definition of $R(n)$ as the algebra of Casimir operators arising in recouplings of $n$ copies of $\mathfrak{su}(1,1)$. These observations provide a natural framework to carry out the derivation by dimensional reduction of the generic superintegrable model on the $(n-1)$ sphere which is invariant under $R(n)$.