The Racah Algebra of Rank 2: Properties, Symmetries and Representation

TL;DR

This paper introduces a new universal definition for the rank-2 Racah algebra, explores its properties and symmetries, and presents a novel representation that simplifies its structure using only one diagonal generator.

Contribution

It provides a universal definition for the rank-2 Racah algebra, analyzes its symmetries, and introduces a simplified representation based on the split basis.

Findings

Verified equivalence of new and existing defining relations

Identified symmetries allowing alternative algebraic formulations

Developed a new representation with only one diagonal generator

Abstract

The goals of this paper are threefold. First, we provide a new ''universal'' definition for the Racah algebra of rank 2 as an extension of the rank-1 Racah algebra where the generators are indexed by subsets and any three disjoint indexing sets define a subalgebra isomorphic to the rank-1 case. With this definition, we explore some of the properties of the algebra including verifying that these natural assumptions are equivalent to other defining relations in the literature. Second, we look at the symmetries of the generators of the rank-2 Racah algebra. Those symmetries allows us to partially make abstraction of the choice of the generators and write relations and properties in a different format. Last, we provide a novel representation of the Racah algebra. This new representation requires only one generator to be diagonal and is based on an expansion of the split basis representation from the rank-1 Racah algebra.