Abstract:
For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation. A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, and analogues are obtained for the three classical theorems of Lie. The problem of constructing global pseudogroups is examined.
Bibliography: 49 titles.
\Bibitem{Kar86}
\by M.~V.~Karasev
\paper Analogues of the objects of Lie group theory for nonlinear Poisson brackets
\jour Math. USSR-Izv.
\yr 1987
\vol 28
\issue 3
\pages 497--527
\mathnet{http://mi.mathnet.ru/eng/im1499}
\crossref{https://doi.org/10.1070/IM1987v028n03ABEH000895}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=854594}
\zmath{https://zbmath.org/?q=an:0624.58007|0608.58023}
Linking options:
https://www.mathnet.ru/eng/im1499
https://doi.org/10.1070/IM1987v028n03ABEH000895
https://www.mathnet.ru/eng/im/v50/i3/p508
This publication is cited in the following 69 articles:
Charles-Michel Marle, Encyclopedia of Mathematical Physics, 2025, 424
Hichem Lassoued, Camille Laurent-Gengoux, “Symplectic resolutions of the quotient of R2 by an infinite symplectic discrete group”, Ann Glob Anal Geom, 67:2 (2025)
Oscar Cosserat, Camille Laurent-Gengoux, Vladimir Salnikov, “Numerical methods in Poisson geometry and their application to mechanics”, Mathematics and Mechanics of Solids, 29:5 (2024), 904
Leonid O. Chekhov, Michael Shapiro, “Log-Canonical Coordinates for Symplectic Groupoid and Cluster Algebras”, Int. Math. Res. Not. IMRN, 2023:11 (2023), 9565–9652
DANIEL ÁLVAREZ, “REDUCTION OF SYMPLECTIC GROUPOIDS AND QUOTIENTS OF QUASI-POISSON MANIFOLDS”, Transformation Groups, 28:4 (2023), 1357
Fabrizio Pugliese, Giovanni Sparano, Luca Vitagliano, “Integrating Nijenhuis structures”, Annali di Matematica, 202:4 (2023), 1907
Fabrizio Pugliese, Giovanni Sparano, Luca Vitagliano, “Multiplicative connections and their Lie theory”, Commun. Contemp. Math., 25:01 (2023)
Oscar Cosserat, “Symplectic groupoids for Poisson integrators”, Journal of Geometry and Physics, 186 (2023), 104751
Vladislav G Kupriyanov, Richard J Szabo, “Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry”, J. Phys. A: Math. Theor., 55:3 (2022), 035201
Thiago Drummond, “Lie–Nijenhuis Bialgebroids”, The Quarterly Journal of Mathematics, 73:3 (2022), 849
Alexander Karabegov, “Lagrangian fields, Calabi functions, and local symplectic groupoids”, Differential Geometry and its Applications, 85 (2022), 101933
Patrick Cabau, Fernand Pelletier, “Integrability on Direct Limits of Banach Manifolds”, Annales de la Faculté des sciences de Toulouse : Mathématiques, 28:5 (2020), 909
Hichem Lassoued, “Dimension preserving resolutions of singular Poisson structures”, Differential Geometry and its Applications, 65 (2019), 212
Richard J. Szabo, “Quantization of Magnetic Poisson Structures”, Fortschritte der Physik, 67:8-9 (2019)
Henrique Bursztyn, Thiago Drummond, “Lie theory of multiplicative tensors”, Math. Ann., 375:3-4 (2019), 1489
Vladislav G. Kupriyanov, Richard J. Szabo, “Symplectic realization of electric charge in fields of monopole distributions”, Phys. Rev. D, 98:4 (2018)
Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, “Remarks on Contact and Jacobi Geometry”, SIGMA, 13 (2017), 059, 22 pp.
L. O. Chekhov, M. Mazzocco, “On a Poisson homogeneous space of bilinear forms with a Poisson–Lie action”, Russian Math. Surveys, 72:6 (2017), 1109–1156
S. Agyo, C. Lei, A. Vourdas, “The groupoid of bifractional transformations”, Journal of Mathematical Physics, 58:5 (2017)
Citation in format AMSBIB
Contact us: