Stokes phenomena, Poisson–Lie groups and quantum groups

Abstract

Let g be a complex semisimple Lie algebra, b±⊂g a pair of opposite Borel subalgebras, and r∈b−⊗b+ the corresponding solution of the classical Yang–Baxter equations. Let G be the simply–connected Poisson–Lie group corresponding to (g,r), H⊂B±⊂G the subgroups with Lie algebras h=b−∩b+ and b±, and G⁎=B+×HB− the Poisson–Lie group dual of G. G–valued Stokes phenomena were used by Boalch [3,4] to give a canonical, analytic linearisation of the Poisson–Lie group 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 Uħg[23]. 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 Uħg is a Stokes matrix for the dynamical KZ equations.