Irregular Riemann-Hilbert correspondence, Alekseev-Meinrenken dynamical r-matrices and Drinfeld twists

TL;DR

This paper links irregular Riemann-Hilbert problems, symplectic geometry, and Drinfeld twists to advance the understanding of the Ginzburg-Weinstein linearization theorem for Lie bialgebras, providing new constructions and interpretations.

Contribution

It introduces a novel connection map construction via irregular Riemann-Hilbert problems and relates it to Drinfeld twists, enriching the geometric and algebraic understanding of Lie bialgebra linearization.

Findings

Connection maps are constructed as solutions to an irregular Riemann-Hilbert problem.

Any solution of the PDEs for gauge transformation is a semiclassical limit of a Drinfeld twist.

New symplectic geometric interpretation of the PDEs and a relation to the Lu-Weinstein symplectic double.

Abstract

In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem for a quasitriangular Lie bialgebra $(\g,r)$. This approach is based on solving a system of PDEs for a gauge transformation between the classical r-matrix $r$ and the Alekseev-Meinrenken dynamical r-matrix. They proved that the semiclassical limit of an admissible Drinfeld twist gives rise to a solution of the PDEs. In this paper, we explain that preferred gauge transformations can be constructed as connection maps for a certain irregular Riemann-Hilbert problem (provided $r$ is the standard classical r-matrix). Along the way, we give a symplectic geometric interpretation of their PDEs, as a symplectic neighborhood version of the Ginzburg-Weinstein linearization theorem. Our construction is based on earlier works by Boalch. We then prove that for semisimple Lie algebra $\g$, any solution of the PDEs for the gauge transformation is the semiclassical limit of an admissible Drinfeld twist. As byproducts, we find a surprising relation between the connection maps and Drinfeld twists as well as a new description of the Lu-Weinstein symplectic double.