Description
As observed by many authors, the theory of finite-dimensional Poisson
algebraic groups and of their quantizations has a natural interpretation
in terms of 2d TFTs equipped with a pair of transverse boundary
conditions. There are then various dualities in this theory which are
reflected in symmetries of the associated TFT. In this talk we'll give
an overview of this approach with an eye toward applications in
low-dimensional topology. One of our main motivation was to understand
the work of Alekseev--Enriquez--Torrossian and of Bar-Natan--Dancso
which shows a surprising connection between the Kashiwara-Vergne
conjecture and the Duflo isomorphism in Lie theory, and the braid groups
actions on character varieties of punctured discs.