On the Casimir of the group ISL(n,R) and its algebraic decomposition

TL;DR

This paper derives an explicit tensor expression for the Casimir operator of the inhomogeneous group ISL(n,R), facilitating particle classification in affine gravitational theories and decomposing the Casimir into those of its subgroups.

Contribution

It provides a new explicit tensor form for the Casimir of ISL(n,R) and demonstrates its decomposition into Casimirs of its little groups, aiding representation theory.

Findings

Explicit tensor expression for the Casimir operator of ISL(n,R)

Decomposition of the Casimir into subgroups' Casimirs

Application to particle classification in gravitational gauge theories

Abstract

In this paper, an explicit expression for the Casimir operator (or the Casimir invariant) of the inhomogeneous group ISL(n,R) in its enveloping algebra is proposed, which using contractions of the tenso- rial indices of the generating operators P^{rho} and E^{nu}{mu} may be presented in the following (slightly more comprehensible as equation (1)) form. The Casimir is obtained by symmetrizing this expression. This tensor form is useful in the classification of particles in affine gravitational gauge theories; such as that based on ISL(4,R). It is also proven that the Casimir of ISL(n,R) can be decomposed in terms of the Casimirs of its little groups, a key point in the posterior construction of its irreducible representations.