Towards a characterization of exact symplectic Lie algebras $\frak{g}$ in terms of the invariants for the coadjoint representation

Abstract

We prove that for any known Lie algebra $\frak{g}$ having none invariants for the coadjoint representation, the absence of invariants is equivalent to the existence of a left invariant exact symplectic structure on the corresponding Lie group $G$. We also show that a nontrivial generalized Casimir invariant constitutes an obstruction for the exactness of a symplectic form, and provide solid arguments to conjecture that a Lie algebra is endowed with an exact symplectic form if and only if all invariants for the coadjoint representation are trivial. We moreover develop a practical criterion that allows to deduce the existence of such a symplectic form on a Lie algebra from the shape of the antidiagonal entries of the associated commutator matrix. In an appendix the classification of Lie algebras satisfying $\mathcal{N}(\frak{g})=0$ in low dimensions is given in tabular form, and their exact symplectic structure is given in terms of the Maurer-Cartan equations.