The Racah algebra as a commutant and Howe duality

TL;DR

This paper explores the Racah algebra's role as a commutant and its connection to Howe duality, revealing new insights into its symmetry properties and relations to superintegrable models and the Racah problem.

Contribution

It establishes a novel link between the Racah algebra, Howe duality, and the symmetry algebra of superintegrable models on the 2-sphere.

Findings

Racah algebra identified as a commutant in oscillator representations

Connection between Racah algebra and Howe duality elucidated

Insights into the Racah problem and superintegrable models provided

Abstract

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the $2$-sphere. It is further identified as the commutant of the $\mathfrak{o}(2) \oplus \mathfrak{o}(2) \oplus \mathfrak{o}(2)$ subalgebra of $\mathfrak{o}(6)$ in oscillator representations of the universal algebra of the latter. How this observation relates to the $\mathfrak{su}(1,1)$ Racah problem and the superintegrable model on the $2$-sphere is discussed on the basis of the Howe duality associated to the pair $\big(\mathfrak{o}(6)$, $\mathfrak{su}(1,1)\big)$.