Obtainment of internal labelling operators as broken Casimir operators by means of contractions related to reduction chains in semisimple Lie algebras

TL;DR

This paper demonstrates how contractions related to reduction chains in semisimple Lie algebras can generate internal labelling operators from broken Casimir operators, aiding in solving the missing label problem.

Contribution

It introduces a method to derive missing label operators from contracted Casimir operators using homogeneous polynomial decompositions.

Findings

Contraction induces a decomposition of Casimir operators into homogeneous polynomials.

Additional mutually commuting label operators can be obtained from these polynomials.

The method helps solve the missing label problem when contractions alone are insufficient.

Abstract

We show that the In\"on\"u-Wigner contraction naturally associated to a reduction chain $\frak{s}\supset \frak{s}^{\prime}$ of semisimple Lie algebras induces a decomposition of the Casimir operators into homogeneous polynomials, the terms of which can be used to obtain additional mutually commuting missing label operators for this reduction. The adjunction of these scalars that are no more invariants of the contraction allow to solve the missing label problem for those reductions where the contraction provides an insufficient number of labelling operators.