The structure of the invariants of perfect Lie algebras

Abstract

The structure of the invariants of perfect Lie algebras with nontrivial centre is analysed. It is shown that if the radical r of a semidirect sum s −→⊕Rr of a semisimple and a nilpotent Lie algebra has a one-dimensional centre, then the defining representation R contains a copy of the trivial representation D0 of s. Using this fact, a criterion can be deduced to eliminate variables and to characterize the Casimir operators of an algebra g by means of certain subrepresentations R � of R. For rank 1 Levi subalgebras s, all representations leading to perfect Lie algebras with a radical isomorphic to a Heisenberg Lie algebra are determined. We prove that the number of Casimir operators of such an algebra is fixed for any dimension, and moreover that they can be explicitly computed by means of determinantal formulae obtained from the brackets of the algebra. For rank n � 2 Levi part a stabilization result of this nature is also given