# Catherine Meusburger (U Erlangen): Introduction to Poisson Lie-groups

**Lecture I March 27, 2017Lecture II March 28, 2017Lecture III March 29, 2017Lecture IV March 31, 2017 **

**Abstract: **A Poisson-Lie group is a Lie group that is also a Poisson manifold in

such a way that the multiplication is a Poisson map.

On the Lie algebra level, this implies that the dual vector space of its

Lie algebra also has a Lie algebra structure, and the two Lie algebra

structures satisfy a compatibility condition.

This is called a Lie bialgebra and can be viewed as the infinitesimal

counterpart of a quantum group. Hence, we can interpret Lie-bialgebras

as the infinitesimal counterparts and Poisson-Lie groups as the

classical counterparts of quantum groups.

I explain these relations and then discuss Poisson actions of

Poisson-Lie groups on Poisson manifolds. I explain why these structures

can be expected to appear in gauge theory. If there is time, I also

cover Drinfeld's classification of Poisson homogeneous spaces, i.e.

Poisson manifolds with transitive Poisson actions of Poisson-Lie groups.