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

TL;DR

This paper proves that quasi-Poisson manifolds with certain structures can be quantized, extending previous results and providing new proofs for classical cases, with applications to dynamical r-matrices.

Contribution

It generalizes the quantization of quasi-Poisson manifolds and introduces a generalized formality theorem, extending known results to broader classes of manifolds.

Findings

Quantization of all quasi-Poisson $( ext{g}, Z)$-manifolds.

New proof of the equivariant formality theorem for Poisson manifolds.

Quantization of modified classical dynamical $r$-matrices over abelian bases.

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