Application of the Gel'fand Matrix Method to the Missing Label Problem in Classical Kinematical Lie Algebras

TL;DR

This paper applies the Gel'fand matrix method to compute invariants and address the missing label problem in (3+1)-dimensional classical kinematical Lie algebras, enhancing understanding of their structure.

Contribution

It demonstrates how the Gel'fand matrix method can be used to find Casimir operators and solve the missing label problem for kinematical Lie algebras.

Findings

Matrices constructed yield invariants via characteristic polynomials

Reductions on matrices assist in identifying missing operators

Method applicable to contractions of simple Lie algebras

Abstract

We briefly review a matrix based method to compute the Casimir operators of Lie algebras, mainly certain type of contractions of simple Lie algebras. The versatility of the method is illustrated by constructing matrices whose characteristic polynomials provide the invariants of the kinematical algebras in (3+1)-dimensions. Moreover it is shown, also for kinematical algebras, how some reductions on these matrices are useful for determining the missing operators in the missing label problem (MLP).