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 $M$.
Categories: math-ph math.DG math.FA math.MP