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Mat. Zametki, 2002, Volume 72, Issue 1, Pages 11–34 (Mi mz400)  

This article is cited in 41 scientific papers (total in 41 papers)

Compatible Poisson Brackets on Lie Algebras

A. V. Bolsinova, A. V. Borisovb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b M. V. Lomonosov Moscow State University

Abstract: We discuss the relationship between the representation of an integrable system as an $L$-$A$-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.


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English version:
Mathematical Notes, 2002, 72:1, 10–30

Bibliographic databases:

UDC: 517.9
Received: 01.10.2000

Citation: A. V. Bolsinov, A. V. Borisov, “Compatible Poisson Brackets on Lie Algebras”, Mat. Zametki, 72:1 (2002), 11–34; Math. Notes, 72:1 (2002), 10–30

Citation in format AMSBIB
\by A.~V.~Bolsinov, A.~V.~Borisov
\paper Compatible Poisson Brackets on Lie Algebras
\jour Mat. Zametki
\yr 2002
\vol 72
\issue 1
\pages 11--34
\jour Math. Notes
\yr 2002
\vol 72
\issue 1
\pages 10--30

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    This publication is cited in the following articles:
    1. I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and Integrable Equations of the Principal Chiral Model Type”, Funct. Anal. Appl., 36:3 (2002), 172–181  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. I. Z. Golubchik, V. V. Sokolov, “Factorization of the Loop Algebra and Integrable Toplike Systems”, Theoret. and Math. Phys., 141:1 (2004), 1329–1347  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    3. A. V. Tsiganov, “On isomorphism of integrable cases of the Euler equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$”, J. Math. Sci. (N. Y.), 136:1 (2006), 3641–3647  mathnet  crossref  mathscinet  zmath  elib
    4. Bolsinov A., “Integrable Geodesic Flows on Riemannian Manifolds: Construction and Obstructions”, Proceedings of the Workshop on Contemporary Geometry and Related Topics, eds. Bokan N., Djoric M., Fomenko A., Rakic Z., Wess J., World Scientific Publ Co Pte Ltd, 2004, 57–103  crossref  mathscinet  zmath  isi
    5. Golubchik, IZ, “Factorization of the loop algebras and compatible Lie brackets”, Journal of Nonlinear Mathematical Physics, 12 (2005), 343  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. L. G. Rybnikov, “The Argument Shift Method and the Gaudin Model”, Funct. Anal. Appl., 40:3 (2006), 188–199  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and the Yang–Baxter Equation”, Theoret. and Math. Phys., 146:2 (2006), 159–169  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. Fiorani, E, “Noncommutative integrability on noncompact invariant manifolds”, Journal of Physics A-Mathematical and General, 39:45 (2006), 14035  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    9. Odesskii, A, “Algebraic structures connected with pairs of compatible associative algebras”, International Mathematics Research Notices, 2006, 43734  mathscinet  zmath  isi  elib
    10. Odesskii, AV, “Integrable matrix equations related to pairs of compatible associative algebras”, Journal of Physics A-Mathematical and General, 39:40 (2006), 12447  crossref  mathscinet  zmath  isi  scopus  scopus
    11. Odesskii, AV, “Compatible Lie brackets related to elliptic curve”, Journal of Mathematical Physics, 47:1 (2006), 013506  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    12. A. V. Tsiganov, “Compatible Lie–Poisson brackets on the Lie algebras $e(3)$ and $so(4)$”, Theoret. and Math. Phys., 151:1 (2007), 459–473  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    13. Damianou, PA, “Integrable hierarchies and the modular class”, Annales de l Institut Fourier, 58:1 (2008), 107  crossref  mathscinet  zmath  isi  scopus  scopus
    14. Kostko, AL, “On the bi-Hamiltonian structures for the Goryachev-Chaplygin top”, Regular & Chaotic Dynamics, 13:1 (2008), 38  mathscinet  zmath  adsnasa  isi
    15. Bolsinov, AV, “Bi-Hamiltonian structures and singularities of integrable systems”, Regular & Chaotic Dynamics, 14:4–5 (2009), 431  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    16. Bloch, AM, “A Class of Integrable Flows on the Space of Symmetric Matrices”, Communications in Mathematical Physics, 290:2 (2009), 399  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    17. A. V. Belyaev, “Asymptotic behaviour of singular points of solutions of the problem of heavy $n$-dimensional body motion in the Lagrange case”, Sb. Math., 202:11 (2011), 1617–1635  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    18. Tsiganov A.V., “On Bi-Integrable Natural Hamiltonian Systems on Riemannian Manifolds”, J. Nonlinear Math. Phys., 18:2 (2011), 245–268  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    19. Campoamor-Stursberg R., Musso F., “Two-Body Homogeneous Rational Gaudin Models and the Missing Label Problem”, J. Phys. A-Math. Theor., 46:33 (2013), 335201  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    20. Alina Dobrogowska, Anatol Odzijewicz, “Integrable Systems Related to Deformed $\mathfrak{so}(5)$”, SIGMA, 10 (2014), 056, 18 pp.  mathnet  crossref  mathscinet
    21. Dobrogowska A., Golinski T., “Lie Bundle on the Space of Deformed Skew-Symmetric Matrices”, J. Math. Phys., 55:11 (2014), 113504  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    22. Izosimov A., “Stability of Relative Equilibria of Multidimensional Rigid Body”, Nonlinearity, 27:6 (2014), 1419–1443  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    23. Pumei Zhang, “Algebraic Properties of Compatible Poisson Brackets”, Regul. Chaotic Dyn., 19:3 (2014), 267–288  mathnet  crossref  mathscinet
    24. Panasyuk A., “Compatible Lie Brackets: Towards a Classification”, J. Lie Theory, 24:2 (2014), 561–623  mathscinet  zmath  isi
    25. Andrey V. Tsiganov, “Simultaneous Separation for the Neumann and Chaplygin Systems”, Regul. Chaotic Dyn., 20:1 (2015), 74–93  mathnet  crossref  mathscinet  zmath
    26. Dobrogowska A., “R-Matrix, Lax pair, and Multiparameter Decompositions of Lie Algebras”, J. Math. Phys., 56:11 (2015), 113508  crossref  mathscinet  zmath  isi  scopus  scopus
    27. Wu Ming-Zhong, Bai Cheng-Ming, “Compatible Lie Bialgebras”, Commun. Theor. Phys., 63:6 (2015), 653–664  crossref  mathscinet  zmath  isi  scopus  scopus
    28. Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Note on Free Symmetric Rigid Body Motion”, Regul. Chaotic Dyn., 20:3 (2015), 293–308  mathnet  crossref  mathscinet  zmath  adsnasa
    29. Guha P., “Nonholonomic Deformation of Coupled and Supersymmetric KdV Equations and Euler-Poincaré-Suslov Method”, Rev. Math. Phys., 27:4 (2015), 1550011  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    30. A. V. Bolsinov, “Argument shift method and sectional operators: applications to differential geometry”, J. Math. Sci., 225:4 (2017), 536–554  mathnet  crossref  mathscinet  elib
    31. Pavel E. Ryabov, Andrej A. Oshemkov, Sergei V. Sokolov, “The Integrable Case of Adler – van Moerbeke. Discriminant Set and Bifurcation Diagram”, Regul. Chaotic Dyn., 21:5 (2016), 581–592  mathnet  crossref  mathscinet  zmath
    32. P. E. Ryabov, E. O. Biryucheva, “Diskriminantnoe mnozhestvo i bifurkatsionnaya diagramma integriruemogo sluchaya M. Adlera i P. van Merbeke”, Nelineinaya dinam., 12:4 (2016), 633–650  mathnet  crossref  elib
    33. Bolsinov A., “Singularities of Bi-Hamiltonian Systems and Stability Analysis”: Bolsinov, A MoralesRuiz, JJ Zung, NT, Geometry and Dynamics of Integrable Systems, Adv. Courses Math CRM Barc., Advanced Courses in Mathematics Crm Barcelona, Birkhauser Verlag Ag, 2016, 35–84  crossref  mathscinet  isi
    34. Daniel J. F. Fox, “Symmetries of the Space of Linear Symplectic Connections”, SIGMA, 13 (2017), 002, 30 pp.  mathnet  crossref
    35. S. V. Sokolov, “Integriruemyi sluchai Adlera–van Mërbeke. Vizualizatsiya bifurkatsii torov Liuvillya”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:4 (2017), 532–539  mathnet  crossref  elib
    36. Dobrogowska A., Jakimowicz G., “Tangent Lifts of Bi-Hamiltonian Structures”, J. Math. Phys., 58:8 (2017), 083505  crossref  mathscinet  zmath  isi  scopus  scopus
    37. Lazureanu C., “On a Hamilton-Poisson Approach of the Maxwell-Bloch Equations With a Control”, Math. Phys. Anal. Geom., 20:3 (2017), 20  crossref  mathscinet  isi  scopus  scopus
    38. Lazureanu C., “Hamilton-Poisson Realizations of the Integrable Deformations of the Rikitake System”, Adv. Math. Phys., 2017, 4596951  crossref  mathscinet  isi  scopus  scopus
    39. Ryabov P.E., “Explicit Integration of the System of Invariant Relations For the Case of M. Adler and P. Van Moerbeke”, Dokl. Math., 95:1 (2017), 17–20  crossref  mathscinet  zmath  isi  scopus  scopus
    40. Konyaev A.Yu., “Completeness of Some Commutative Subalgebras Associated With Nijenhuis Operators on Lie Algebras”, Dokl. Math., 97:2 (2018), 137–139  crossref  zmath  isi  scopus  scopus
    41. Borisov A., Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520  mathscinet  isi
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