Integrable hierarchies  and the modular class

Pantelis A. Damianou; Rui Loja Fernandes

Displaying similar documents to “Integrable hierarchies and the modular class”

Poisson manifolds, Lie algebroids, modular classes: a survey.

Kosmann-Schwarzbach, Yvette (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

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Modular vector fields and Batalin-Vilkovisky algebras

Yvette Kosmann-Schwarzbach (2000)

Banach Center Publications

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We show that a modular class arises from the existence of two generating operators for a Batalin-Vilkovisky algebra. In particular, for every triangular Lie bialgebroid (A,P) such that its top exterior power is a trivial line bundle, there is a section of the vector bundle A whose $d_{P}$ -cohomology class is well-defined. We give simple proofs of its properties. The modular class of an orientable Poisson manifold is an example. We analyse the relationships between generating operators of the...

The modular class of a Poisson map

Raquel Caseiro, Rui Loja Fernandes (2013)

Annales de l’institut Fourier

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We introduce the modular class of a Poisson map. We look at several examples and we use the modular classes of Poisson maps to study the behavior of the modular class of a Poisson manifold under different kinds of reduction. We also discuss their symplectic groupoid version, which lives in groupoid cohomology.

Normal forms of vector fields on Poisson manifolds

Philippe Monnier, Nguyen Tien Zung (2006)

Annales mathématiques Blaise Pascal

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We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

Multi-Hamiltonian structures on Beauville's integrable system and its variant.

Inoue, Rei, Konishi, Yukiko (2007)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

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Some Remarks on Dirac Structures and Poisson Reductions

Zhang-Ju Liu (2000)

Banach Center Publications

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Dirac structures are characterized in terms of their characteristic pairs defined in this note and then Poisson reductions are discussed from the point of view of Dirac structures.

Nambu-Poisson Tensors on Lie Groups

Nobutada Nakanishi (2000)

Banach Center Publications

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First as an application of the local structure theorem for Nambu-Poisson tensors, we characterize them in terms of differential forms. Secondly left invariant Nambu-Poisson tensors on Lie groups are considered.

Poisson structures on $ℝ^{2 N}$ having only two symplectic leaves: the origin and the rest

S. Zakrzewski (2000)

Banach Center Publications

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Examples of Poisson structures with isolated non-symplectic points are constructed from classical r-matrices.

Poisson Lie groups and their relations to quantum groups

Janusz Grabowski (1995)

Banach Center Publications

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The notion of Poisson Lie group (sometimes called Poisson Drinfel'd group) was first introduced by Drinfel'd [1] and studied by Semenov-Tian-Shansky [7] to understand the Hamiltonian structure of the group of dressing transformations of a completely integrable system. The Poisson Lie groups play an important role in the mathematical theories of quantization and in nonlinear integrable equations. The aim of our lecture is to point out the naturality of this notion and to present basic...