Quantization of $r-Z$-quasi-Poisson manifolds and related modified classical dynamical $r$-matrices

Abstract

Le $X$ be a $C^\infty$-manifold and $\g$ be a finite dimensional Lie algebra acting freely on $X$. Let $r \in \ve^2(\g)$ be such that $Z=[r,r] \in \ve^3(\g)^\g$. In this paper we prove that every quasi-Poisson $(\g,Z)$-manifold can be quantized. This is a generalization of the existence of a twist quantization of coboundary Lie bialgebras (\cite{EH}) in the case $X=G$ (where $G$ is the simply connected Lie group corresponding to $\g$). We deduce our result from a generalized formality theorem. In the case Z=0, we get a new proof of the existence of (equivariant) formality theorem and so (equivariant) quantization of Poisson manifold ({\it cf.} \cite{Ko,Do}). As a consequence of our results, we get quantization of modified classical dynamical $r$-matrices over abelian bases in the reductive case