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

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dc.creatorCampoamor-Stursberg, Rutwig
dc.date2003-01-06
dc.date.accessioned2026-07-07T04:29:42Z
dc.date.available2026-07-07T04:29:42Z
dc.descriptionWe 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.
dc.description21 pages, 5 tables
dc.identifierhttps://arxiv.org/abs/math-ph/0301004
dc.identifierhttp://arxiv.org/abs/math-ph/0301004
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/57257
dc.subjectMathematical Physics
dc.subject17B10, 81R05
dc.titleTowards a characterization of exact symplectic Lie algebras $\frak{g}$ in terms of the invariants for the coadjoint representation
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