Drinfeld Correspondence in Infinite Dimensions

TL;DR

This paper extends the Drinfeld correspondence between Poisson Lie groups and Lie bialgebras to infinite-dimensional regular Lie groups modeled on convenient vector spaces, including loop and diffeomorphism groups.

Contribution

It generalizes the Drinfeld correspondence to infinite-dimensional settings, focusing on regular Lie groups modeled on nuclear Fréchet and Silva spaces.

Findings

Established Drinfeld correspondence for infinite-dimensional Lie groups.

Extended the correspondence to loop groups and diffeomorphism groups.

Provided examples involving smooth and analytic structures.

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 $M$.