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Regul. Chaotic Dyn., 2014, Volume 19, Issue 3, Pages 267–288 (Mi rcd146)  

This article is cited in 6 scientific papers (total in 6 papers)

Algebraic Properties of Compatible Poisson Brackets

Pumei Zhangab

a China University of Political Science and Law, 25 Xitucheng Lu, Haidian District, Beijing, 100088, China
b School of Mathematics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom

Abstract: We discuss algebraic properties of a pencil generated by two compatible Poisson tensors $\mathcal A(x)$ and $\mathcal B(x)$. From the algebraic viewpoint this amounts to studying the properties of a pair of skew-symmetric bilinear forms $\mathcal A$ and $\mathcal B$ defined on a finite-dimensional vector space. We describe the Lie group $G_{\mathcal P}$ of linear automorphisms of the pencil $\mathcal P = \{\mathcal A + \lambda\mathcal B\}$. In particular, we obtain an explicit formula for the dimension of $G_{\mathcal P}$ and discuss some other algebraic properties such as solvability and Levi – Malcev decomposition.

Keywords: compatible Poisson brackets, Jordan–Kronecker decomposition, pencils of skew symmetric matrices, bi-Hamiltonian systems

Funding Agency
This paper is supported by Program for Young Innovative Research Team in China University of Political Science and Law 2014CXTD06.


DOI: https://doi.org/10.1134/S1560354714030010

References: PDF file   HTML file

Bibliographic databases:

MSC: 15A21, 15A22, 17B45, 17B80, 37J35, 53D17
Received: 31.08.2013
Accepted:26.03.2014
Language:

Citation: Pumei Zhang, “Algebraic Properties of Compatible Poisson Brackets”, Regul. Chaotic Dyn., 19:3 (2014), 267–288

Citation in format AMSBIB
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\by Pumei~Zhang
\paper Algebraic Properties of Compatible Poisson Brackets
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 3
\pages 267--288
\mathnet{http://mi.mathnet.ru/rcd146}
\crossref{https://doi.org/10.1134/S1560354714030010}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3215689}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000337051600001}


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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. S. Rosemann, K. Schoebel, “Open problems in the theory of finite-dimensional integrable systems and related fields”, J. Geom. Phys., 87 (2015), 396–414  crossref  mathscinet  zmath  isi  scopus
    2. F. Dopico, F. Uhlig, “Computing matrix symmetrizers, part 2: New methods using eigendata and linear means; a comparison”, Linear Alg. Appl., 504 (2016), 590–622  crossref  mathscinet  zmath  isi  scopus
    3. A. V. Bolsinov, P. Zhang, “Jordan-Kronecker invariants of finite-dimensional Lie algebras”, Transform. Groups, 21:1 (2016), 51–86  crossref  mathscinet  zmath  isi  scopus
    4. A. Bolsinov, “Singularities of bi-Hamiltonian systems and stability analysis”: A. Bolsinov, J. J. Morales-Ruiz, Nguyen Tien Zung, Geometry and Dynamics of Integrable Systems, Advanced Courses in Mathematics – CRM Barcelona, Birkhauser Verlag Ag, 2016, 35–84  crossref  mathscinet  isi
    5. A. V. Bolsinov, A. M. Izosimov, D. M. Tsonev, “Finite-dimensional integrable systems: a collection of research problems”, J. Geom. Phys., 115 (2017), 2–15  crossref  mathscinet  zmath  isi  scopus
    6. A. Bolsinov, V. S. Matveev, E. Miranda, S. Tabachnikov, “Open problems, questions and challenges in finite-dimensional integrable systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170430  crossref  mathscinet  zmath  isi  scopus
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