Poisson and Symplectic structures, Hamiltonian action, momentum and reduction

Vladimir Roubtsov (1) , Denys Dutykh (2, 3, 4)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques
2 LAMA - Laboratoire de Mathématiques
3 INSMI-CNRS - Institut National des Sciences Mathématiques et de leurs Interactions - CNRS Mathématiques
4 USMB [Université de Savoie] [Université de Chambéry] - Université Savoie Mont Blanc

This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisla-2019, Poland and written down by the second author. As the title suggests, the material covered here includes the Poisson and symplectic structures (Poisson manifolds, Poisson bi-vectors and Poisson brackets), group actions and orbits (infinitesimal action, stabilizers and adjoint representations), moment maps, Poisson and Hamiltonian actions. Finally, the phase space reduction is also discussed. The very last section introduces the Poisson-Lie structures along with some related notions. This text represents a brief review of a well-known material citing standard references for more details. The exposition is concise but pedagogical. The Authors believe that it will be useful as an introductory exposition for students interested in this specific topic.

Mots clés: Hamiltonian action; Poisson and Symplectic structures; momentum and reduction

Domaines: Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Mathématiques générales [math.GM]; Systèmes Solubles et Intégrables [nlin.SI]

Vladimir Roubtsov, Denys Dutykh. Poisson and Symplectic structures, Hamiltonian action, momentum and reduction. 2020. ⟨hal-02522156⟩