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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Kosmann-Schwarzbach, Yvette (2008)
SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]
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Pantelis A. Damianou, Rui Loja Fernandes (2008)
Annales de l’institut Fourier
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It is well-known that the Poisson-Nijenhuis manifolds, introduced by Kosmann-Schwarzbach and Magri form the appropriate setting for studying many classical integrable hierarchies. In order to define the hierarchy, one usually specifies in addition to the Poisson-Nijenhuis manifold a bi-hamiltonian vector field. In this paper we show that to every Poisson-Nijenhuis manifold one can associate a canonical vector field (no extra choices are involved!) which under an appropriate assumption...
Ping Xu (2003)
Annales scientifiques de l'École Normale Supérieure
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Alan Weinstein (2000)
Banach Center Publications
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Henrique Bursztyn, Olga Radko (2003)
Annales de l’institut Fourier
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We study gauge transformations of Dirac structures and the relationship between gauge and Morita equivalences of Poisson manifolds. We describe how the symplectic structure of a symplectic groupoid is affected by a gauge transformation of the Poisson structure on its identity section, and prove that gauge-equivalent integrable Poisson structures are Morita equivalent. As an example, we study certain generic sets of Poisson structures on Riemann surfaces: we find complete gauge-equivalence...
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 -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...
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.
Vaisman, I. (1999)
Acta Mathematica Universitatis Comenianae. New Series
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Dimitri Gurevich, Pavel Saponov (2011)
Banach Center Publications
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We consider Poisson pencils, each generated by a linear Poisson-Lie bracket and a quadratic Poisson bracket corresponding to a so-called Reflection Equation Algebra. We show that any bracket from such a Poisson pencil (and consequently, the whole pencil) can be restricted to any generic leaf of the Poisson-Lie bracket. We realize a quantization of these Poisson pencils (restricted or not) in the framework of braided affine geometry. Also, we introduce super-analogs of all these Poisson...