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Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 3, Pages 508–538 (Mi izv1499)  

This article is cited in 35 scientific papers (total in 35 papers)

Analogues of the objects of Lie group theory for nonlinear Poisson brackets

M. V. Karasev


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.

Full text: PDF file (3783 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1987, 28:3, 497–527

Bibliographic databases:

UDC: 517.9
MSC: Primary 58F05; Secondary 22E70, 81C25
Received: 22.02.1984
Revised: 20.01.1986

Citation: M. V. Karasev, “Analogues of the objects of Lie group theory for nonlinear Poisson brackets”, Izv. Akad. Nauk SSSR Ser. Mat., 50:3 (1986), 508–538; Math. USSR-Izv., 28:3 (1987), 497–527

Citation in format AMSBIB
\Bibitem{Kar86}
\by M.~V.~Karasev
\paper Analogues of the objects of Lie group theory for nonlinear Poisson brackets
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1986
\vol 50
\issue 3
\pages 508--538
\mathnet{http://mi.mathnet.ru/izv1499}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=854594}
\zmath{https://zbmath.org/?q=an:0624.58007|0608.58023}
\transl
\jour Math. USSR-Izv.
\yr 1987
\vol 28
\issue 3
\pages 497--527
\crossref{https://doi.org/10.1070/IM1987v028n03ABEH000895}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. M. Vorob'ev, M. V. Karasev, “Poisson manifolds and the schouten bracket”, Funct. Anal. Appl., 22:1 (1988), 1–9  mathnet  crossref  mathscinet  zmath  isi
    2. A WEINSTEIN, “Connections of Berry and Hannay type for moving Lagrangian submanifolds”, Advances in Mathematics, 82:2 (1990), 133  crossref
    3. I. Szymczak, S. Zakrzewski, “Quantum deformations of the Heisenberg group obtained by geometric quantization”, Journal of Geometry and Physics, 7:4 (1990), 553  crossref
    4. K. H. Bhaskara, K. Rama, “Quadratic Poisson structures”, J Math Phys (N Y ), 32:9 (1991), 2319  crossref  mathscinet  zmath  adsnasa
    5. Ping Xu, “Morita equivalence of Poisson manifolds”, Comm Math Phys, 142:3 (1991), 493  crossref  mathscinet  zmath
    6. Zhang-Ju Liu, Ping Xu, “On quadratic Poisson structures”, Lett Math Phys, 26:1 (1992), 33  crossref  mathscinet  isi
    7. A. Cabras, A.M. Vinogradov, “Extensions of the Poisson bracket to differential forms and multi-vector fields”, Journal of Geometry and Physics, 9:1 (1992), 75  crossref
    8. Dimitri Kusnezov, “On the evaluation of ensemble averages”, Physics Letters B, 289:3-4 (1992), 395  crossref
    9. S Zakrzewski, J Phys A Math Gen, 28:24 (1995), 7347  crossref  mathscinet  zmath  adsnasa
    10. F. Alcalde Cuesta, G. Hector, “Intégration Symplectique des Variétés de Poisson Régulières”, Isr J Math, 90:1-3 (1995), 125  crossref  mathscinet  zmath  isi
    11. S Zakrzewski, J Phys A Math Gen, 29:8 (1996), 1841  crossref  mathscinet  zmath  adsnasa
    12. Manuel de León, Juan C. Marrero, Edith Padrón, “On the geometric quantization of Jacobi manifolds”, J Math Phys (N Y ), 38:12 (1997), 6185  crossref  mathscinet  zmath  elib
    13. S Zakrzewski, J Phys A Math Gen, 30:18 (1997), 6535  crossref  mathscinet  zmath  adsnasa
    14. Janusz Grabowski, “Z-Graded Extensions of Poisson Brackets”, Rev. Math. Phys, 09:01 (1997), 1  crossref
    15. M Karasev, “Advances in quantization: quantum tensors, explicit star-products, and restriction to irreducible leaves”, Differential Geometry and its Applications, 9:1-2 (1998), 89  crossref  elib
    16. A Weinstein, “Poisson geometry”, Differential Geometry and its Applications, 9:1-2 (1998), 213  crossref  elib
    17. Fani Petalidou, “Sur la symplectisation de structures bihamiltoniennes”, Bulletin des Sciences Mathématiques, 124:4 (2000), 255  crossref
    18. V. A. DOLGUSHEV, “SKLYANIN BRACKET AND DEFORMATION OF THE CALOGERO–MOSER SYSTEM”, Mod. Phys. Lett. A, 16:26 (2001), 1711  crossref
    19. ALBERTO S. CATTANEO, GIOVANNI FELDER, “Poisson SIGMA MODELS AND DEFORMATION QUANTIZATION”, Mod. Phys. Lett. A, 16:04n06 (2001), 179  crossref
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    23. A. I. Bondal, “Symplectic Groupoids Related to Poisson–Lie Groups”, Proc. Steklov Inst. Math., 246 (2004), 34–53  mathnet  mathscinet  zmath
    24. A. I. Bondal, “A symplectic groupoid of triangular bilinear forms and the braid group”, Izv. Math., 68:4 (2004), 659–708  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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    27. Alexander V. Karabegov, “Fedosov’s formal symplectic groupoids and contravariant connections”, Journal of Geometry and Physics, 56:10 (2006), 1985  crossref
    28. M. A. Olshanetsky, “Elliptic hydrodynamics and quadratic algebras of vector fields on a torus”, Theoret. and Math. Phys., 150:3 (2007), 301–314  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    30. Yvette Kosmann-Schwarzbach, “Poisson Manifolds, Lie Algebroids, Modular Classes: a Survey”, SIGMA, 4 (2008), 005, 30 pp.  mathnet  crossref  mathscinet  zmath
    31. Alexander Karabegov, “Infinitesimal Deformations of a Formal Symplectic Groupoid”, Lett Math Phys, 2011  crossref
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    34. Andrew James Bruce, Katarzyna Grabowska, Janusz Grabowski, “Remarks on Contact and Jacobi Geometry”, SIGMA, 13 (2017), 059, 22 pp.  mathnet  crossref
    35. 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  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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