Abstract
This manuscript is essentially a collection of lecture notes which were given by the first author at the Summer School Wisła–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.
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Notes
- 1.
Here we mean that the bi-vector π is non-degenerate, i.e.π is invertible when it is seen as a banal matrix.
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Acknowledgements
D.D. was supported by the Laboratory of Mathematics (LAMA UMR #5127) and the University Savoie Mont Blanc to attend the meeting in Wisła. The work of D.D. has been also supported by the French National Research Agency, through Investments for Future Program (ref. ANR− 18 −EURE− 0016—Solar Academy). V.R. acknowledges a partial support of the project IPaDEGAN (H2020-MSCA-RISE-2017), Grant Number 778010, and of the Russian Foundation for Basic Research under the Grants RFBR 18 − 01 − 00461 and 16 − 51 − 53034 − 716 GFEN. Both authors would like to thank the Baltic Mathematical Institute for organizing this scientific event and the anonymous Referee who helped us to improve the presentation indicating some shortcomings and misprints.
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Roubtsov, V., Dutykh, D. (2021). Poisson and Symplectic Structures, Hamiltonian Action, Momentum and Reduction. In: Ulan, M., Schneider, E. (eds) Differential Geometry, Differential Equations, and Mathematical Physics. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-63253-3_1
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