The course is an introduction to Poisson geometry. In the first part, we shall introduce the basic notions related to Poisson geometry and we shall prove some basic results as the Weinstein s plitting theorem and the existence of symplectic realization.
In the last part of the course, we shall introduce the basic notions about Lie algebroids and Lie groupoids and explain theire relationship with Poisson structures.
Syllabus
1st lecture: Poisson brackets. Linear Poisson brackets on the dual of a Lie algebra.
2nd lecture: Examples of Poisson manifolds, Hamiltonian vector fields, Poisson morphisms.
3rd lecture: Schouten bracket, Poisson bivectors.
4th lecture: Weinstein splitting theorem, symplectic leaves.
5th lecture: Poisson-Lie structures, Lie bi-algebras.
6th lecture: The generalized tangent bundle, Dirac structures.
7th lecture: Poisson sprays, symplectic realizations.
8th lecture: Lie groupoids, examples.
9th lecture: Lie algebroid associated to a Lie groupoid, Lie algebroids as linear Poisson structures. Poisson bivector as Lie algebroids.
10th lecture: Lie II and Lie III for Lie algebroids.
References
1.Crainic, Marius and Fernandes, Rui Loja and Marcut¸, Ioan. Lectures on Poisson geometry. Vol. 217. American Mathematical Soc., 2021.
2.Da Silva, Ana Cannas, and Alan Weinstein. Geometric models for noncommutative algebras. Vol. 10. American Mathematical Soc., 1999.
3. Dufour, Jean-Paul and Zung, Nguyen Tien. Poisson structures and their normal forms. Vol. 242. Springer Science Business Media, 2006.
4 Karasev, Mihail Vladimirovi, and Viktor Pavlovi Maslov. Nonlinear Poisson brackets: geometry and quantization. Vol. 119. American Mathematical Soc., 2012.
5. Meinrenken, Eckhard. Poisson geometry from a Dirac perspective. Letters in Mathematical Physics 108.3 (2018): 447-498.