ar X iv : m at h - ph / 0 30 10 04 v 1 6 J an 2 00 3 Towards a characterization of exact symplectic Lie algebras g in terms of the invariants for the coadjoint representation

We prove that for any known Lie algebra 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 commuta-tor matrix. In an appendix the classification of Lie algebras satisfying N (g) = 0 in low dimensions is given in tabular form, and their exact symplectic structure is given in terms of the Maurer-Cartan equations.