Formality theorem for Lie bialgebras and quantization of coboundary r-matrices

TL;DR

This paper proves a formality theorem linking Lie bialgebras and their quantizations, establishing a new $L_$-morphism and deriving a quantization of coboundary r-matrices.

Contribution

It introduces a novel $L_$-morphism between Lie algebra complexes and tensor algebras, enabling the quantization of coboundary r-matrices.

Findings

Existence of an $L_$-morphism between $C(g)$ and $TU$.

Construction of a quantization $R$ for coboundary r-matrices.

Application of formality morphism to derive quantization results.

Abstract

Let $(g,\delta_\hbar)$ be a Lie bialgebra. Let $(U_\hbar(g),\Delta_\hbar)$ a quantization of $(g,\delta_\hbar)$ through Etingof-Kazhdan functor. We prove the existence of a $L_\infty$-morphism between the Lie algebra $C(\g)=\Lambda(g)$ and the tensor algebra $TU=T(U_\hbar(g)[-1])$ with Lie algebra structure given by the Gerstenhaber bracket. When $(g,\delta_\hbar,r)$ is a coboundary Lie bialgebra, we deduce from the formality morphism the existence of a quantization $R$ of $r$.