Stokes phenomena, Poisson-Lie groups and quantum groups

TL;DR

This paper explores the connection between Stokes phenomena, Poisson-Lie groups, and quantum groups, demonstrating how classical structures emerge as limits of quantum constructions and linking R-matrices to Stokes matrices.

Contribution

It shows that the classical G-valued Stokes phenomena can be derived as a semiclassical limit of the quantum Stokes phenomena related to the Drinfeld-Jimbo quantum group.

Findings

Classical Stokes phenomena are the semiclassical limits of quantum Stokes phenomena.

The R-matrix of the quantum group is identified as a Stokes matrix for dynamical KZ equations.

A new link between Poisson-Lie groups and quantum groups via Stokes phenomena is established.

Abstract

Let g be a complex semisimple Lie algebra, G the simply-connected Poisson-Lie group corresponding to g, and G* its dual. G-valued Stokes phenomena were used by Boalch [Bo1,Bo2] to give a canonical, analytic linearisation of the Poisson structure on G*. Ug-valued Stokes phenomena were used by the first author to construct a twist killing the KZ associator, and therefore give a transcendental construction of the Drinfeld-Jimbo quantum group U_hg (arXiv:1601.04076). In the present paper, we show that the former construction can be obtained as semiclassical limit of the latter. Along the way, we also show that the R-matrix of U_hg is a Stokes matrix for the dynamical KZ equations.