Integrability of quotients in Poisson and Dirac geometry

Abstract

We study the integrability of Poisson and Dirac structures that arise from quotient constructions. From our results we deduce several classical results as well as new applications. We study the integrability of quasi-Poisson quotients in full generality recovering, in particular, the integrability of quotients of Poisson manifolds by Poisson actions. We also give explicit constructions of Lie groupoids integrating two interesting families of geometric structures: (i) a special class of Poisson homogeneous spaces of symplectic groupoids integrating Poisson groups and (ii) Dirac homogeneous spaces.

Quotients of multiplicative forms and Poisson reduction

“…M Ã is isomorphic to the pullback algebroid. Some results in [1] can be used to provide a criterion for q to admit an integration to a simple quotient of a Lie groupoid, as follows. Recall that, since q is of pullback type, there is a Lie algebroid injection I A ≡ (ρ| K ) −1 : Ker(T q M ) → A over id M , where the involutive distribution Ker(T q M ) ⊂ T M is endowed with its natural algebroid structure.…”

“…1.1]: since, in these cases, q is of pullback type, the hypothesis of q admitting an integration by a simple groupoid quotient is replaced by the criterion recalled in Example 3.4 and one can show that, under this hypothesis, source-connectedness is not necessary to ensure ω is q-basic. ((G, ω) is called a "q M -admissible presymplectic integration" in [1].) This refined criterion can be used proceduce to integrations of a variety of Poisson manifolds obtained as quotients, including Poisson homogeneous spaces, see [1,7].…”

“…((G, ω) is called a "q M -admissible presymplectic integration" in [1].) This refined criterion can be used proceduce to integrations of a variety of Poisson manifolds obtained as quotients, including Poisson homogeneous spaces, see [1,7]. We stress that their criterion to produce the groupoid simple quotient q integrating the pullback type quotient q is non-trivial (see Example 3.4 and the cited references) and is independent of the topics covered in the present paper.…”

“…If the foliation defined by D is simple with underlying simple quotient q M : M → M , then E := Ann M (D) induces Poisson quotient data as above such that q * A ≃ T * π M where π is the Poisson structure obtained from π via standard coisotropic reduction. (In the particular case in which π♯ | Ann M (T M ) is injective, the underlying simple quotient is of pullback type and a specialized integration treatment can be found in [1,Prop. 3.17].)…”

“…Several particular cases of this integration problem appear in the literature and, from these, is clear the the source-simply-connected integration does not provide a solution in general and adhoc techniques have to be considered. A particular case in which the quotient q is of "pullback type" (see Remark 2.16) and k = 2 can be treated by the techniques introduced by D. Álvarez [1] and the outcome is a complete characterization of the underlying integrability conditions (see Example 3.4). For general q, as far as the authors know, the integrability problem has to be studied in a case-by-case basis and a similar discussion applies to the quotient map behind Ker(ω).…”