On the unicity of braidings of quasitriangular Lie bialgebras

TL;DR

This paper proves that all braidings arising from quantizations of quasitriangular Lie bialgebras are identical to the Weinstein-Xu braiding, and it characterizes lifts of classical r-matrices with universal formulas.

Contribution

It establishes the unicity of the braiding associated with quantizations and provides a universal formula for lifts of classical r-matrices using co-Hochschild cohomology.

Findings

All quantization-induced braidings coincide with Weinstein-Xu braiding.

Existence and uniqueness of lifts of classical r-matrices are proven.

Lifts can be expressed via universal formulas.

Abstract

Any quantization of a quasitriangular Lie bialgebra g gives rise to a braiding of the dual Poisson-Lie formal group G^*. We show that this braiding always coincides with the Weinstein-Xu braiding. We also define the lifts of the classical r-matrix r as certain functions on G^* x G^*, prove their existence and uniqueness using co-Hochschild cohomology arguments and show that the lift can be expressed in terms of r by universal formulas.