Manin pairs and moment maps revisited

TL;DR

This paper revisits the theory of quasi-Poisson spaces with moment maps, establishing a bijective correspondence between different categories and providing new constructions and examples, including from moduli spaces.

Contribution

It introduces a Lifting Theorem linking categories of quasi-Poisson spaces with different moment map values, enabling new constructions and examples.

Findings

Established a bijective correspondence between categories of quasi-Poisson spaces.

Provided new constructions of fusion and conjugation for these spaces.

Presented new examples from moduli spaces.

Abstract

The notion of quasi-Poisson $G$-spaces with $D/G$-valued moment maps was introduced by Alekseev and Kosmann-Schwarzbach in 1999. Our main result is a \emph{Lifting Theorem}, establishing a bijective correspondence between the categories of quasi-Poisson $G$-spaces with $D/G$-valued moment maps and of quasi-Poisson $G\times G$-spaces with $D$-valued moment maps. Using this result, we give simple constructions of fusion and conjugation for these spaces, and new examples coming from moduli spaces.