Manin pairs and moment maps revisited

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 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× 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.