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Many relevant applications of group theoretical methods to physical problemsare related, in some manner, to classification schemes by means of symmetrygroups. In these schemes, irreducible representations of a Lie group have to bedecomposed according to some subgroup, thus providing a labeling of states interms of the subgroup. This is the typical situation in nuclear physics, whereHamiltonians are described as functions of the Casimir operators of the groupsinvolved in some reduction chain. This further gives an effective method todeduce the corresponding energy formulae. It is not however infrequent that theembedding of a Lie group as subgroup of a larger symmetry group is notcanonical, hence leading to different branching rules of representations and toan insufficient number of labeling operators.  The objective of this work is to review the actual developments of thecontraction method, including also reduction chains with respect to abelian(Cartan) subalgebras. The decomposition of the invariants of the coadjointrepresentation provide natural classes of labeling operators for thecorresponding embeddings, which are of interest for various specific problemsranging from nuclear physics to the group theoretical foundations of theperiodic table of chemical elements.

Embeddings of Lie algebras, contractions and the state labelling problem