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Some remarks on dressing transformations. (English) Zbl 0653.58012

The space of one-forms on a Poisson manifold (P,\(\pi)\) has the structure of a Lie algebra under the bracket \[ \{w_ 1,w_ 2\}=d(\pi (w_ 1,w_ 2))-i_{\pi w_ 1}dw_ 2+i_{\pi w_ 2}dw_ 1. \] A Poisson Lie group is a Lie group endowed with a Poisson structure for which the multiplication mapping is a Poisson map. The following theorem appearing in M. V. Karasev, Izv. Akad. Nauk SSSR, Ser. Mat. 50, No.3, 508-538 (1986; Zbl 0608.58023) is given here a simple proof: The right (left) invariant one-forms on a Poisson Lie group form a subalgebra. The corresponding Lie algebra structure on the dual of the Lie algebra of the Poisson Lie group is equal to the one given by linearizing the Poisson structure on G at the identity.
Reviewer: T.Ratiu

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0608.58023
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