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Dressing transformations and Poisson group actions. (English) Zbl 0674.58038

A Lie group G is called a Poisson Lie group if a Poisson bracket is fixed on G such that the multiplication \(G\times G\to G\) is a Poisson mapping, the space \(G\times G\) being equipped with the Poisson structure. Let M be a Poisson manifold. An action \(G\times M\to M\) is called Poisson action if it is a Poisson mapping, the space \(G\times M\) being equipped with the product Poisson structure.
The main result of this article is as follows: dressing transformation groups define a Poisson group action. The Poisson properties of dressing transformations in soliton theory are explained.
Reviewer: B.V.Loginov

MSC:

37C80 Symmetries, equivariant dynamical systems (MSC2010)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J90 Applications of PDEs on manifolds
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