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Momentum mappings and Poisson cohomology. (English) Zbl 0867.58028

The by now standard theory about the equivariant momentum mapping associated to the action of a semisimple Lie group on a symplectic manifold, becomes a highly non-trivial issue in the context of Poisson actions on Poisson manifolds \((P,\pi)\). The paper starts with an introduction to the geometry of Poisson manifolds, which includes an example of explicit calculation of Poisson cohomology. In Section 3, an important construction is outlined of a Lie derivative operation \({\mathcal L}_\alpha\) with respect to 1-forms, by which the spaces of forms and multi-vector fields acquire the structure of \(\Omega^1(P)\)-modules. It is shown that \({\mathcal L}_\alpha\) gives rise to flows, in general different from the flow of \(\pi^\#\alpha\), which have much better features with respect to Poisson properties. One of the main results of Section 4 is a criterion for uniqueness of an equivariant momentum mapping. As for existence of such a mapping, the author focusses on actions which are “cotangential”, meaning that the generators come from 1-forms, and arrives at complete answers when the action preserves \(\pi\). It is further shown that the induced action on the Poisson cohomology is trivial, provided that the action on \(P\) is cotangential and the group is compact or semisimple. Section 5 is about equivariant cotangent lifts.
Reviewer: W.Sarlet (Gent)

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B56 Cohomology of Lie (super)algebras
17B99 Lie algebras and Lie superalgebras
57S15 Compact Lie groups of differentiable transformations
53D50 Geometric quantization
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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