Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

In a nutshell, a Poisson-Lie group is a Lie group with a certain type of compatible Poisson structure. From a physical viewpoint, the study of Poisson-Lie groups was initially motivated by the analysis of the Hamiltonian structure of the group of so-called dressing transformations arising in the inspection of relevant classes of integrable systems. The aim of this talk is to introduce the theory of Poisson-Lie groups and their dressing transformations. After recalling several basic results on Poisson manifolds, I will characterise Poisson-Lie groups in different equivalent ways, I will study the properties of their related Lie algebras (which leads to the theory of Lie bialgebras), and I will analyse the main properties of dressing transformations.