Internal labelling operators and contractions of Lie algebras

TL;DR

This paper explores how Inönü-Wigner contractions can solve the missing label problem in Lie algebra reductions, linking label operators to Casimir operators of contracted algebras and providing bounds for invariants.

Contribution

It introduces a new approach connecting missing label operators with Casimir operators via contractions, offering insights into invariants of affine Lie algebras.

Findings

Missing label operators relate to Casimir operators of contracted algebras.

Contraction-based method provides bounds for invariants of affine Lie algebras.

Available labeling operators are not all equivalent.

Abstract

We analyze under which conditions the missing label problem associated to a reduction chain $\frak{s}^{\prime}\subset \frak{s}$ of (simple) Lie algebras can be completely solved by means of an In\"on\"u-Wigner contraction $\frak{g}$ naturally related to the embedding. This provides a new interpretation of the missing label operators in terms of the Casimir operators of the contracted algebra, and shows that the available labeling operators are not completely equivalent. Further, the procedure is used to obtain upper bounds for the number of invariants of affine Lie algebras arising as contractions of semisimple algebras.