Braiding structures on formal Poisson groups and classical solutions of the QYBE

If g is a quasitriangular Lie bialgebra, the formal Poisson group F[[g^*]] can be given a braiding structure. This was achieved by Weinstein and Xu using purely geometrical means, and independently by the authors by means of quantum groups. In this paper we compare these two approaches. First, we show that the braidings they produce share several similar properties (in particular, the construction is functorial); secondly, in the simplest case (G = SL_2) they do coincide. The question then rises of whether they are always the same this is positively answered in a separate paper

Quantization of Γ-Lie bialgebras

AbstractWe introduce the notion of Γ-Lie bialgebras, where Γ is a group. These objects give rise to cocommutative co-Poisson bialgebras, for which we construct quantization functors. This enlarges the class of co-Poisson algebras for which a quantization is known. Our result relies on our earlier work, where we showed that twists of Lie bialgebras can be quantized; we complement this work by studying the behavior of this quantization under compositions of twists

An ℏ-adic valuation property of universal R-matrices

AbstractWe prove that if Uℏ(g) is a quasitriangular QUE algebra with universal R-matrix R, and Oℏ(G∗) is the quantized function algebra sitting inside Uℏ(g), then ℏlog(R) belongs to the tensor square Oℏ(G∗)⊗̄Oℏ(G∗). This gives another proof of the results of Gavarini and Halbout, saying that R normalizes Oℏ(G∗)⊗̄Oℏ(G∗) and therefore induces a braiding of the formal group G∗ (in the sense of Weinstein and Xu, or Reshetikhin)