In this chapter we discuss Poisson–Lie groups and their infinitesimal counterparts Lie bialgebras. A Poisson–Lie group is a Lie group G which is equipped with a Poisson structure π, having the property that the product map on G is a morphism of Poisson manifolds. A Lie bialgebra is a Lie algebra which is at the same time a Lie coalgebra, the algebra and coalgebra structures satisfying some compatibility relation. We show that there is a functor which associates to every Poisson–Lie group a Lie bialgebra and that every finite-dimensional Lie bialgebra is the Lie bialgebra of some Poisson–Lie group, which can be chosen to be connected and simply connected. Using dressing actions, we shortly discuss the symplectic leaves of Poisson–Lie groups.
KeywordsPoisson Structure Linear Poisson Structure Unique Poisson Structure
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