Manin Pairs and Moment Maps

Abstract

A Lie group G in a group pair (D, G), integrating a Lie algebra g in a Manin pair (d, g), has a quasi-Poisson structure. We define the quasi-Poisson actions of such Lie groups G, that generalize the Poisson actions of Poisson Lie groups. We define and study the moment maps for those quasi-Poisson actions which are quasi-hamiltonian. These moment maps take values in the homogeneous space D/G. We prove an analogue of the hamiltonian reduction theorem for quasi-Poisson group actions, and we study the symplectic leaves of the orbit spaces of quasi-hamiltonian spaces.

The chiral WZNW phase space as a quasi-Poisson space

“…The final notion to be recalled is that of a quasi-Poisson space [12]. This is a manifold M which is equipped with a bivector P M and on which a (left) action of a connected quasiPoisson-Lie group G is given subject to some conditions.…”

“…The global objects corresponding to Lie quasi-bialgebras (G,F ,φ) are the quasi-Poisson-Lie groups [8,12]. Such a Lie group with Lie algebra G is equipped with a multiplicative bivector that corresponds toF and satisfies certain conditions involvingφ.…”

“…Note that the requirements (14) and (15) ensure that the quasi-Poisson bracket can be restricted to the space of invariant functions where it becomes a genuine Poisson bracket, which is also invariant under twisting [12].…”

“…We first recall from [6,12] that a Manin quasi-triple is a Lie algebra D with a non-degenerate, invariant scalar product | and decomposition D = G ⊕ H, where G and H are maximal isotropic subspaces and G is also a Lie subalgebra in D. (D, G, H) becomes a Manin triple if H is a Lie subalgebra, too. Choosing a basis {e i } i = 1, 2, .…”

“…Apparently, the notion of a classical symmetry action of a quasi-Poisson-Lie group itself has only been defined very recently [12,13]. The purpose of this letter is to point out that the expectation just alluded to is indeed correct, since the group G equipped with the so called canonical quasi-Poisson-Lie structure, which is the classical limit of A h (G), acts as a symmetry in the sense of [12] on the chiral WZNW quasi-Poisson space. We shall see that the simplest quasi-Poisson structure on M chir for which this holds can still be defined by (2) but withr = 0.…”