Quotients of multiplicative forms and Poisson reduction

Abstract

In this paper we study quotients of Lie algebroids and groupoids endowed with compatible differential forms. We identify Lie theoretic conditions under which such forms become basic and characterize the induced forms on the quotients. We apply these results to describe generalized quotient and reduction processes for (twisted) Poisson and Dirac structures, as well as to their integration by (twisted, pre-)symplectic groupoids. In particular, we recover and generalize several known results concerning Poisson reduction.

Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds

“…Remark 3.7. As we shall see next, the integrability of the quotient Poisson structure is controlled by the integrability of this Lie algebroid structure on C. It is immediate that the inclusion C ↪ T * S is a closed IM 2-form on C, so we see that quasi-Poisson reduction fits within the general framework of Poisson reduction described in [11].…”

“…In the case of a Poisson manifold (M, π), the identity on T * M is a closed IM 2-form. Closed IM 2-forms are the basic infrastructure necessary for performing Poisson reduction as we shall see below, see [11] for a general discussion. At the Lie groupoid level, a closed IM 2-form induces a closed multiplicative 2-form: let G ⇉ M be a Lie groupoid and take [45,25] if it is endowed with a symplectic form which is multiplicative.…”