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.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2003
- DOI:
- 10.48550/arXiv.math-ph/0301004
- arXiv:
- arXiv:math-ph/0301004
- Bibcode:
- 2003math.ph...1004C
- Keywords:
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- Mathematical Physics;
- Mathematics - Mathematical Physics;
- 17B10;
- 81R05
- E-Print:
- 21 pages, 5 tables