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 Fr\'{e}chet 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 $M$.