A. V. Tsiganov, “Compatible Lie–Poisson brackets on the Lie algebras $e(3)$ and $so(4)$”, TMF, 151:1 (2007), 26–43; Theoret. and Math. Phys., 151:1 (2007), 459

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Teoreticheskaya i Matematicheskaya Fizika, 2007, Volume 151, Number 1, Pages 26–43
DOI: https://doi.org/10.4213/tmf6009 (Mi tmf6009)

This article is cited in 10 scientific papers (total in 10 papers)

Compatible Lie–Poisson brackets on the Lie algebras

e (3)

$e(3)$ and

s o (4)

$so(4)$

A. V. Tsiganov

St. Petersburg State University, Faculty of Physics

Full-text PDF (493 kB) Citations (10)

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DOI: https://doi.org/10.4213/tmf6009

Abstract: We completely classify the compatible Lie–Poisson brackets on the dual spaces of the Lie algebras

e (3)

$e(3)$ and

s o (4)

$so(4)$ . The corresponding bi-Hamiltonian systems are the spinning tops corresponding to the classical cases of integrability of the Euler equations, the Kirchhoff equations, and the Poincaré–Zhukovskii equations.

Keywords: integrable system, bi-Hamiltonian manifold, Lie–Poisson tensor.

Received: 03.07.2006
Revised: 12.10.2006

English version:
Theoretical and Mathematical Physics, 2007, Volume 151, Issue 1, Pages 459–473
DOI: https://doi.org/10.1007/s11232-007-0034-z

Bibliographic databases:

Language: Russian

Citation: A. V. Tsiganov, “Compatible Lie–Poisson brackets on the Lie algebras

e (3)

$e(3)$ and

s o (4)

$so(4)$ ”, TMF, 151:1 (2007), 26–43; Theoret. and Math. Phys., 151:1 (2007), 459–473

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf6009

https://doi.org/10.4213/tmf6009

https://www.mathnet.ru/eng/tmf/v151/i1/p26

This publication is cited in the following 10 articles:

Jiefeng Liu, Yunhe Sheng, Chengming Bai, “Maurer-Cartan characterizations and cohomologies of compatible Lie algebras”, Sci. China Math., 66:6 (2023), 1177
Andrey V. Tsiganov, “Reduction of Divisors and the Clebsch System”, Regul. Chaotic Dyn., 27:3 (2022), 307–319
Alina Dobrogowska, Anatol Odzijewicz, “Integrable Systems Related to Deformed $\mathfrak{so}(5)$ ”, SIGMA, 10 (2014), 056, 18 pp.
Andrey Tsiganov, “Poisson structures for two nonholonomic systems with partially reduced symmetries”, Journal of Geometric Mechanics, 6:3 (2014), 417
Tsiganov A.V., “New Variables of Separation for Particular Case of the Kowalevski Top”, Regular & Chaotic Dynamics, 15:6 (2010), 659–669
A. V. Tsyganov, “O novom razdelenii peremennykh dlya chastnogo sluchaya volchka Kovalevskoi”, Nelineinaya dinam., 6:3 (2010), 639–652
Tsiganov AV, “The Poisson bracket compatible with the classical reflection equation algebra”, Regular & Chaotic Dynamics, 13:3 (2008), 191–203
Tsiganov AV, “On bi-Hamiltonian structure of some integrable systems on so*(4)”, Journal of Nonlinear Mathematical Physics, 15:2 (2008), 171–185
Tsiganov, AV, “Separation of variables for a pair of integrable systems on so*(4)”, Doklady Mathematics, 76:3 (2007), 839
Tsiganov, AV, “A family of the Poisson brackets compatible with the Sklyanin bracket”, Journal of Physics A-Mathematical and Theoretical, 40:18 (2007), 4803

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

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