Drinfeld Correspondence in Infinite Dimensions

Abstract

In this article, we establish the Drinfeld correspondence between Poisson Lie groups and their infinitesimal counterparts, Lie bialgebras, in the infinite-dimensional setting. Specifically, we extend this correspondence to regular Lie groups modeled on convenient vector spaces, with a particular focus on those modeled on nuclear Fréchet and nuclear Silva spaces. Important examples of interest include the smooth loop group $C^{\infty }({\mathbb{S}}^1,G)$ and the analytic loop group $C^{\omega }({\mathbb{S}}^1,G)$ of a 1-connected real Lie group $G$, as well as $\widetilde{{\mathrm{Diff}}^{\infty }(M{)}_0}$ and $\widetilde{{\mathrm{Diff}}^{\omega }(M{)}_0}$ -- the universal covering groups of the identity components of the groups of smooth and real-analytic diffeomorphisms of a compact manifold $M$.