The structure of the invariants of perfect Lie algebras

Abstract

Upper bounds for the number N(g) of Casimir operators of perfect Lie algebras g with nontrivial Levi decomposition are obtained, and in particular the existence of nontrivial invariants is proved. It is shown that for high-ranked representations R the Casimir operators of the semidirect sum s −→⊕ R(deg R)L1 of a semisimple Lie algebra s and an Abelian Lie algebra (deg R)L1 of dimension equal to the degree of R are completely determined by the representation R, which also allows the analysis of the invariants of subalgebras which extend to operators of the total algebra. In particular, for the adjoint representation of a semisimple Lie algebra the Casimir operators of s −→⊕ ad(s)(dims)L1 can be explicitly constructed from the Casimir operators of the Levi part s.