Compatible Poisson brackets on Lie algebras
type: Item
instance of: scholarly article
title: Compatible Poisson brackets on Lie algebras
zbMATH Open document ID: 1042.37041
DOI: 10.1023/A:1019856702638
author: A. V. Borisov
author: A. V. Bolsinov
published in: Mathematical Notes
publication date: 2003-06-15T00:00:00Z
review text: The authors discuss the relationship between the representation of an integrable system via a Lax pair with a spectral parameter and the existence of a bi-Hamiltonian representation. After reviewing some well-known methods of producing bi-Hamiltonian systems, the authors state the following result: Let \(\{~,~\}_{\lambda}\) be a family of Poisson brackets on a linear vector space and \(v\) a vector field, such that: (i) Almost all brackets in the family are isomorphic to the dual of a semisimple Lie algebra, and (ii) \(v\) is Hamiltonian with respect to all brackets from the family. Then \(v\) admits a Lax representation with a sprectral parameter. The authors illustrate this result by considering some well-known examples of completely integrable systems: geodesic flows of left-invariant metrics on Lie groups, the Zhukovskii-Volterra system describing the inertial motion of a balanced gyroscope, the Kowaleski top, etc.
reviewed by: Rui Loja Fernandes
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 17B63
Mathematics Subject Classification ID: 53D17
Mathematics Subject Classification ID: 37K30
zbMATH DE Number: 1928258
zbMATH Keywords: compatible Poisson brackets
zbMATH Keywords: Lax pairs
zbMATH Keywords: bi-Hamiltonian representation
zbMATH Keywords: completely integrable systems
MaRDI profile type: Publication
full work available at URL: a:1019856702638
OpenAlex ID: W2150594106
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description: scientific article; zbMATH DE number 1928258