All Finite (Anti)Hermitian Irreducible Representations of the de Sitter and Anti-de Sitter Lie Algebras and Their Lorentz Structure

TL;DR

This paper classifies all finite-dimensional irreducible representations of de Sitter and anti-de Sitter Lie algebras with Hermitian or anti-Hermitian generators, detailing their structure, matrix elements, and Casimir invariants, contrasting with the infinite-dimensional unitary case.

Contribution

It provides a complete classification and explicit matrix elements of finite-dimensional irreducible representations of de Sitter and anti-de Sitter groups based on Lorentz sub-algebra structures.

Findings

Identified 17 irreps with dimensions less than 105.

Derived explicit Casimir invariants in terms of two positive integers.

Connected finite-dimensional irreps to known infinite-dimensional unitary representations.

Abstract

Because of the importance of unitarity in quantum physics, work on the representations of the de Sitter group has focussed on the unitary case, which necessarily means infinite dimensional matrices for this non-compact group. Here we address the finite dimensional representations resulting from the requirement that the Lie algebra generators are either Hermitian or anti-Hermitian. The complete classification of all such irreducible representations is found and their matrix elements specified. These irreducible representations (irreps) are based on backbones defined as the homogeneous Lorentz sub-algebra and consisting of direct sums of the finite irreps of the homogeneous Lorentz algebra (HLA). All these irreps have been characterised by a diagrammatic depiction conveniently labelled by two positive integers m/n where m,n lie in (1,2,3,..) excepting only m=n=1. The two Casimir invariants have been given explicitly in terms of the two positive integers, m and n, equations (11.11a,b), and shown to reproduce the result familiar from the study of the infinite dimensional, unitary representations -C1=p(p+1)+(q+1)(q-2) and -C2=p(p+1)q(q-1) but with a reduced set of possible p,q values. There are 17 irreps with dimensions less than 105, having dimensions 4, 5, 10, 14, 16, 20, 30, 35(x2), 40, 55, 56, 64, 80, 81, 84 and 91. The corresponding irreps of the anti-de Sitter group follow immediately from the replacement of the 4-momentum operators V{\mu} by iV{\mu}.