TY - CHAP

T1 - Poisson and Symplectic Structures, Hamiltonian Action, Momentum and Reduction

AU - Roubtsov, Vladimir

AU - Dutykh, Denys

N1 - Funding Information:
Acknowledgments 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.
Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85129211344&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-63253-3_1

DO - 10.1007/978-3-030-63253-3_1

M3 - Chapter

AN - SCOPUS:85129211344

T3 - Tutorials, Schools, and Workshops in the Mathematical Sciences

SP - 1

EP - 29

BT - Tutorials, Schools, and Workshops in the Mathematical Sciences

ER -