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

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Matematicheskie Zametki, 2002, Volume 72, Issue 1, Pages 11–34
DOI: https://doi.org/10.4213/mzm400 (Mi mzm400)

This article is cited in 61 scientific papers (total in 61 papers)

Compatible Poisson Brackets on Lie Algebras

A. V. Bolsinov^a, A. V. Borisov^b

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

Full-text PDF (321 kB) Citations (61)

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DOI: https://doi.org/10.4213/mzm400

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.

Received: 01.10.2000

English version:
Mathematical Notes, 2002, Volume 72, Issue 1, Pages 10–30
DOI: https://doi.org/10.1023/A:1019856702638

Bibliographic databases:

UDC: 517.9

Language: Russian

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

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Linking options:

https://www.mathnet.ru/eng/mzm400

https://doi.org/10.4213/mzm400

https://www.mathnet.ru/eng/mzm/v72/i1/p11

This publication is cited in the following 61 articles:

Taoufik Chtioui, Ripan Saha, “On deformation cohomology of compatible Hom-associative algebras”, Afr. Mat., 36:1 (2025)
Abdenacer Makhlouf, Ripan Saha, “On compatible Leibniz algebras”, J. Algebra Appl., 24:04 (2025)
Taoufik Chtioui, Apurba Das, Sami Mabrouk, “(Co)homology of compatible associative algebras”, Communications in Algebra, 52:2 (2024), 582
Tomasz Goliński, Alice Barbora Tumpach, “Geometry of Integrable Systems Related to the Restricted Grassmannian”, SIGMA, 20 (2024), 104, 18 pp.
Alina Dobrogowska, Grzegorz Jakimowicz, Marzena Szajewska, Trends in Mathematics, Geometric Methods in Physics XXXIX, 2023, 169
Jiefeng Liu, Yunhe Sheng, Chengming Bai, “Maurer-Cartan characterizations and cohomologies of compatible Lie algebras”, Sci. China Math., 66:6 (2023), 1177
Apurba Das, “Compatible L∞-algebras”, Journal of Algebra, 610 (2022), 241
Dobrogowska A., Wojciechowicz K., “Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras”, Symmetry-Basel, 13:8 (2021), 1455
Dinar Ya.I., “Algebraic Classical W-Algebras and Frobenius Manifolds”, Lett. Math. Phys., 111:5 (2021), 115
Lazureanu C., Petrisor C., Hedrea C., “On a Deformed Version of the Two-Disk Dynamo System”, Appl. Mat., 66:3 (2021), 345–372
Dobrogowska A., Jakimowicz G., “Generalization of the Concept of Classical R-Matrix to Lie Algebroids”, J. Geom. Phys., 165 (2021), 104227
A. Yu. Konyaev, “Completeness of commutative Sokolov-Odesskii subalgebras and Nijenhuis operators on $\operatorname{gl}(n)$ ”, Sb. Math., 211:4 (2020), 583–593
Roman-Roy N., “A Summary on Symmetries and Conserved Quantities of Autonomous Hamiltonian Systems”, J. Geom. Mech., 12:3 (2020), 541–551
Dinar Ya., “On Integrability of Transverse Lie-Poisson Structures At Nilpotent Elements”, J. Geom. Phys., 155 (2020), 103690
Borisov A., Mamaev I., “Rigid Body Dynamics”, Rigid Body Dynamics, de Gruyter Studies in Mathematical Physics, 52, Walter de Gruyter Gmbh, 2019, 1–520
Dobrogowska A., Jakimowicz G., Wojciechowicz K., “Bi-Hamiltonian Structures on the Tangent Bundle to a Poisson Manifold”, J. Geom. Symmetry Phys., 52 (2019), 47–66
Wu M., “Double Constructions of Compatible Associative Algebras”, Algebr. Colloq., 26:3 (2019), 479–494
Konyaev A.Yu., “Completeness of Some Commutative Subalgebras Associated With Nijenhuis Operators on Lie Algebras”, Dokl. Math., 97:2 (2018), 137–139
Daniel J. F. Fox, “Symmetries of the Space of Linear Symplectic Connections”, SIGMA, 13 (2017), 002, 30 pp.
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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