XLII Workshop on Geometric Methods in Physics Białystok, 30.6-5.07.2025 XIV School on Geometry and Physics Białystok, 23-27.06.2025

Praful Rahangdale

Poisson Geometry and related Geometric and Algebraic Structures

A Poisson structure is a foliation on a manifold, which partitions it into leaves. These leaves have a symplectic structure, and transverse to the leaves, there is a Lie algebra structure. Thus, Poisson geometry amalgamates foliation theory, symplectic geometry, and Lie theory.

The original motivation for studying Poisson structures comes from classical mechanics and integrable systems. Phase spaces of classical mechanical systems are modeled on Poisson manifolds. The time evolution of the system is given by the flow of the vector field corresponding to a distinguished function called the Hamiltonian of the system. The system is constrained to submanifolds, which are the level sets of functions that Poisson commute with the Hamiltonian. The set of these commuting functions are conserved quantities of the system.

In this talk, we shall first discuss the motivation and important examples of Poisson structures. Then, we will explore the local description of a Poisson manifold according to Weinstein splitting theorem, as a product of a symplectic manifold with another manifold transverse to it, whose Poisson structure vanishes at the point. After introducing symplectic realizations of Poisson manifolds, we will see that any Poisson manifold can be viewed as the quotient of a symplectic manifold. Finally, we will discuss Lie bialgebras, the algebraic counterpart of Poisson Lie groups, and the one-to-one correspondence between connected, simply connected Poisson Lie groups and finite-dimensional Lie bialgebras.