Integrability of Poisson Brackets

By M. Crainic, Rui Loja Fernandes. 2002

56 citations recorded by the source corpus.

We discuss the integration of Poisson brackets, motivated by our recent solution to the integrability problem for general Lie brackets. We give the precise obstructions to integrating Poisson manifolds, describing the integration as a symplectic quotient, in the spirit of the Poisson sigma-model of Cattaneo and Felder. For regular Poisson manifolds we express the obstructions in terms of variations of symplectic areas, improving on results of Alcalde Cuesta and Hector. We apply our results (and our point of view) to decide about the existence of complete symplectic realizations, to the integrability of submanifolds of Poisson manifolds, and to the study of dual pairs, Morita equivalence and reduction.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/91d6ed0bbb3a77457188d46d3baf5a67065bd464

Citations in the corpus

• Lectures on integrability of Lie brackets (2006, unknown): “The integration of Poisson submanifolds (and more general submanifolds) was discussed first by Xu in [63] and then completed by us in [16].”
• Symplectic and Poisson geometry on b-manifolds (2012, unknown): “This was obtained using [CF04, Corollary 14] and as a consequence of the existence of a leafwise symplectic embedding for Z .”