Catherine Meusburger (U Erlangen): Introduction to Poisson Lie-groups
Lecture I March 27, 2017
Lecture II March 28, 2017
Lecture III March 29, 2017
Lecture IV March 31, 2017
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.