Reduced Poisson actions

2000 ◽  
Vol 43 (10) ◽  
pp. 1026-1034 ◽  
Author(s):  
Zhangju Liu ◽  
Qilin Yang
Keyword(s):  
Poisson Actions

scholarly journals Poisson actions and scattering theory for integrable systems

1998 ◽  
Vol 4 (1) ◽  
pp. 315-402 ◽  
Author(s):  
Chuu-Lian Terng ◽  
Karen Uhlenbeck
Keyword(s):  
Integrable Systems ◽  
Scattering Theory ◽  
Poisson Actions

Poisson actions

1997 ◽  
Vol 86 (2) ◽  
pp. 261-304 ◽  
Author(s):  
Jiang-Hua Lu
Keyword(s):  
Poisson Actions

MOMENTUM MAPPINGS AND POISSON COHOMOLOGY

1996 ◽  
Vol 07 (03) ◽  
pp. 329-358 ◽  
Author(s):  
VIKTOR L. GINZBURG
Keyword(s):  
Poisson Structure ◽  
Poisson Manifold ◽  
Existence Problem ◽  
Necessary And Sufficient Condition ◽  
Momentum Mapping ◽  
Lie Algebra Cohomology ◽  
Poisson Cohomology ◽  
Poisson Actions ◽  
Poisson Lie Groups ◽  
Necessary And Sufficient

We analyze the question of existence and uniqueness of equivariant momentum mappings for Poisson actions of Poisson Lie groups. A necessary and sufficient condition for the equivariant momentum mapping to be unique is given. The existence problem is solved under some extra hypotheses, for example, when the action preserves the Poisson structure. In this case, the problem is closely related to the triviality of the induced group action on the Poisson cohomology. This action is shown to be trivial whenever the group is compact or semisimple. Conceptually, these results rely upon a version of “Poisson calculus” developed here to make one-forms on a Poisson manifold induce a “flow” preserving the Poisson structure. In the general case, obstructions to the existence of an infinitesimal version of an equivariant momentum mapping are found. Using Lie algebra cohomology with coefficients in Fréchet modules, we show that the obstructions vanish, and the infinitesimal mapping exists, when the group is compact semisimple. We also prove the rigidity of compact group actions preserving the Poisson structure on a compact manifold and calculate the Poisson cohomology of the Poisson homogeneous space [Formula: see text].

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