Contractions of Lie Algebras and Generalized Casimir Invariants

Abstract

We prove that if g' is a contraction of a Lie algebra g then the number of functionally independent invariants of g' is at least that of g. This allows to obtain some criteria to ensure the non-existence of non-trivial invariants for Lie algebras, as well as to deduce some results on the number of derivations of a Lie algebra. In particular, it is shown that almost any even dimensional solvable complete Lie algebra has only trivial invariants. Moreover, with the contraction formula we determine explicitly the number of invariants of Lie algebras carrying a supplementary structure, such as linear contact or linear forms whose differential is symplectic, without having explicit knowledge on the structure of the contracting algebra. This in particular enables us to construct Lie algebras with non-trivial Levi decomposition and none invariants for the coadjoint representation as deformations of frobeniusian model Lie algebras.