Embeddings of Lie algebras, contractions and the state labelling problem

Many relevant applications of group theoretical methods to physical problems are related, in some manner, to classification schemes by means of symmetry groups. In these schemes, irreducible representations of a Lie group have to be decomposed according to some subgroup, thus providing a labeling of states in terms of the subgroup. This is the typical situation in nuclear physics, where Hamiltonians are described as functions of the Casimir operators of the groups involved in some reduction chain. This further gives an effective method to deduce the corresponding energy formulae. It is not however infrequent that the embedding of a Lie group as subgroup of a larger symmetry group is not canonical, hence leading to different branching rules of representations and to an insufficient number of labeling operators. The objective of this work is to review the actual developments of the contraction method, including also reduction chains with respect to abelian (Cartan) subalgebras. The decomposition of the invariants of the coadjoint representation provide natural classes of labeling operators for the corresponding embeddings, which are of interest for various specific problems ranging from nuclear physics to the group theoretical foundations of the periodic table of chemical elements.