Internal labelling operators and contractions of Lie algebras

Abstract

We analyse under which conditions the missing label problem associated with a reduction chain of (simple) Lie algebras can be completely solved by means of an Inönü–Wigner contraction 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 labelling 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 semi-simple algebras.