Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Drinfeld Correspondence in Infinite Dimensions

Praful Rahangdale

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 $C^ω(\mathbb{S}^{1}, G)$ of a 1-connected real Lie group $G$, as well as $\widetilde{\mathrm{Diff}^{\infty}(M)_0}$ and $\widetilde{\mathrm{Diff}^ω(M)_0}$ -- the universal covering groups of the identity components of the groups of smooth and real-analytic diffeomorphisms of a compact manifold $M$.

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 and the analytic loop group of a 1-connected real Lie group , as well as and -- the universal covering groups of the identity components of the groups of smooth and real-analytic diffeomorphisms of a compact manifold .
Paper Structure (44 sections, 52 theorems, 256 equations, 1 figure)

This paper contains 44 sections, 52 theorems, 256 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Structure of the article
  3. Poisson structures on infinite-dimensional manifolds
  4. Poisson Lie groups and their linearization at the identity
  5. Regular Lie group structure on a semidirect product
  6. Lie bialgebra, Lie-Poisson space and Manin triple
  7. Manin triples from weight space decompositions
  8. Smooth vectors and their weight space decomposition
  9. Weight space decomposition
  10. Rapid decay of Fourier coefficients for smooth vectors
  11. Partial sum convergence for smooth vectors
  12. Analytic vectors and their weight decomposition
  13. $A_{\omega}$ is a Silva space
  14. Weight space decomposition
  15. Convergence of partial sums
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $(G, \mathbb{F}_{\mathfrak{b}}, \pi)$ be a convenient Poisson Lie group with Lie algebra $\mathfrak{g}$. Let $\mathfrak{b} \subseteq \mathfrak{g}'$ be a convenient space such that the canonical map $\operatorname{incl}: \mathfrak{b} \to \mathfrak{g}', \alpha \mapsto \alpha$ is smooth, and the co

Figures (1)

  • Figure 1: Isomorphisms for the infinite-dimensional Serre-Swan theorem.

Theorems & Definitions (135)

  • Theorem 1.1: see Theorem \ref{['D-1']}
  • Theorem 1.2: see Theorem \ref{['D-2']}
  • Theorem 1.3: see Theorem \ref{['D-3']}
  • Theorem 1.4: Equivalence between $\mathbf{PLGrp}_{\text{reg}}^{\text{sc}}$ and $\mathbf{LieBiAlg}_{\text{reg}}$, see Theorem \ref{['equivalence']}
  • Definition 2.1: Weak subbundle
  • Definition 2.2: Poisson structure
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • ...and 125 more