Integrable Systems of Neumann Type

Dobrogowska, Alina; Raţiu, Tudor S.

doi:10.1007/s10884-013-9314-5

Cited by 4 publications

(6 citation statements)

References 15 publications

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“…Those brackets were also used to analyze Clebsh and Neumann systems in [3]. We recommend the articles [5][6][7], where the bracket (2) or some of its modification were used to consider some integrable systems.…”

Section: Introductionmentioning

confidence: 99%

Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Dobrogowska

Wojciechowicz

2021

Symmetry

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We present a new look at the classification of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.

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Section: Introductionmentioning

confidence: 99%

Linear Bundle of Lie Algebras Applied to the Classification of Real Lie Algebras

Dobrogowska

Wojciechowicz

2021

Symmetry

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“…. = a n−1 = 1, we obtain Lie algebra e(n − 1) of Euclidean group with Lie bracket given as a commutator modified by a matrix S. More examples of integrable systems related to Lie algebra A a 1 ,...,a n−1 with standard commutator were considered in [DO12,DR13,DO14].…”

Section: Introductionmentioning

confidence: 99%

Lie bundle on the space of deformed skew-symmetric matrices

Dobrogowska

Goliński

2014

Journal of Mathematical Physics

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We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real numbers $a_1,\ldots,a_{n-1}$. Using isomorphism $\mathcal A_{a_1,\ldots,a_{n-1}}^* \cong L_+$ we introduce a Lie-Poisson structure on the space of upper-triangular matrices $L_+$. In this way we generate hierarchies of Hamilton systems with bihamiltonian structure

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“…Since the case α = 0 was considered in [6] we will not discuss it here. The bi-Hamiltonian systems given by the Lie-Poisson bracket {·, ·} 1,α and the constant Lie-Poisson bracket were studied in [7]. By Magri method [12] it can be shown that Casimir functions of the Poisson bracket {·, ·} ,α :…”

Section: Introductionmentioning

confidence: 99%