Stokes phenomena, quantum groups and Poisson-Lie groups.

Quantum groups have long been known to be related to Conformal Field Theory through the Knizhnik-Zamolodchikov (KZ) equations. I will explain a more recent version of this construction which relies on the dynamical  KZ (DKZ) equations. Unlike their precursors, these have irregular singularities. The corresponding Stokes matrices, which encode the discontinuous change of asymptotics of solutions near singular points then turn out to be the R-matrices of the quantum group.
 
In a parallel development, Boalch constructed the Poisson structure on the dual $G*$ of a complex reductive group $G$ by using Stokes phenomena for the simplest irregular connection on the trivial $G$-bundle over $\mathbb{P}^1$.
 
I will briefly review Boalch’s construction, and show that it can be obtained as a semiclassical limit of the construction of quantum groups from the DKZ equations.