2003/01/06 by Rutwig Campoamor-Stursberg, Campoamor-Stursberg, Rutwig
Mathematics · Physics and Astronomy · #17B10 #81R05 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #math-ph #math.MP #msc:17B10 #msc:81R05
paper · pdf · doi:10.48550/arxiv.math-ph/0301004
21 pages, 5 tables
arxiv created 2003/01/06 · openalex publication_date 2003/01/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that for any known Lie algebra \frakg 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 N(\frakg)=0 in low dimensions is given in tabular form, and their exact symplectic structure is given in terms of the Maurer-Cartan equations.