First Integrals Generated by Pseudosymmetries in Nambu-Poisson Mechanics

Crâşmăreanu, Mircea

doi:10.2991/jnmp.2000.7.2.5

Cited by 41 publications

(38 citation statements)

References 13 publications

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“…In particular, if R is an O-operator of g associated with the representation (ad, g), it is known [22] that there exists a pre-Lie algebra structure on g given by…”

Section: )mentioning

confidence: 99%

Some results on L-dendriform algebras

Bai

Liu

2010

Journal of Geometry and Physics

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We introduce a notion of L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the $\mathcal O$-operators of pre-Lie algebras and the related $S$-equation. As a direct consequence, they provide some explicit solutions of $S$-equation in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce a notion of $\mathcal O$-operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra.Comment: 15 page

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“…In particular, if R is an O-operator of g associated with the representation (ad, g), it is known [22] that there exists a pre-Lie algebra structure on g given by…”

Section: )mentioning

confidence: 99%

Some results on L-dendriform algebras

Bai

Liu

2010

Journal of Geometry and Physics

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“…The two-dimensional system, (1.19) and (1.20), is a particular instance of a class of quadratic systems integrable by the factorisation method of AdlerKostant-Symes and its generalisation [5,6] which have recently been studied from the point of view of their symmetry and singularity properties [10].…”

Section: Second-order Ordinary Differential Equations Invariant Undermentioning

confidence: 99%

Symmetry, Singularities, and Integrability in Complex Dynamics II. Rescaling and Time-Translation in Two-Dimensional Systems

Cotsakis

Flessas

2000

Journal of Mathematical Analysis and Applications

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The explicit integrability of second-order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the Painlevé and singularity properties of the different systems. The transformation contains a parameter and for critical values, intimately related to the possession of the Painlevé property in the parent second-order equation, one finds a difference from the generic behaviour. This study is a prelude to a full discussion of the class of transformations which preserve the Painlevé property in the construction of quadratic systems from scalar nth-order ordinary differential equations invariant under time invariant under time translation and rescaling.

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“…Moreover, they have very close relations with many problems in mathematical physics. For example, they appear as an underlying structure of those Lie algebras that possess a phase space ( [15][16][17][18], thus they form a natural category from the point of view of classical and quantum mechanics) and there is a close relation between them and classical Yang-Baxter equation [9,10,12].…”

Section: Introductionmentioning

confidence: 99%