Lie bialgebras, Poisson Lie groups and dressing transformations. (English) Zbl 1078.37517

Kosmann-Schwarzbach, Yvette (ed.) et al., Integrability of nonlinear systems. Proceedings of the CIMPA school, Pondicherry Univ., India, January 8–26, 1996. Berlin: Springer (ISBN 3-540-63353-7). Lect. Notes Phys. 495, 104-170 (1997).
Summary: An essential ingredient of this theory and its applications to integrable systems is the notion of a Poisson action (of a Poisson Lie group on a Poisson manifold). It is a new concept which reduces to that of a Hamiltonian action when the Poisson structure on the Lie group vanishes. It was necessary to introduce such a generalization of Hamiltonian actions in order to account for the properties of the dressing transformations, under the ‘hidden symmetry group’, of fields satisfying a zero-curvature equation. There are naturally defined actions of any Poisson Lie group on the dual Lie group, and conversely, and these are Poisson actions. (We give a one-line proof of this fact in Appendix 2, using the Poisson calculus.) In the case of a Poisson Lie group defined by a factorizable \(R\)-matrix, the explicit formulae for these dressing actions coincide with the dressing of fields that are solutions of zero-curvature equations. There is a notion of momentum mapping for Poisson actions, and in this case it coincides with the monodromy matrix of the linear system. This establishes a connection between soliton equations which admit a zero-curvature representation (the compatibility condition for an auxiliary linear problem) in which the wave function takes values in a group and the theory of Poisson Lie groups.
For the entire collection see [Zbl 0879.00077].


37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
17B62 Lie bialgebras; Lie coalgebras
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
53D17 Poisson manifolds; Poisson groupoids and algebroids