Finite-Dimensional Integration of Poisson Manifolds

Definitions and mathematical statement

Let $(M,\Pi)$ be an integrable Poisson manifold. A Poisson bivector defines a bracket $\{f,g\}=\Pi({\,\mathrm{d}} f,{\,\mathrm{d}} g)$ satisfying the Jacobi identity; its cotangent Lie algebroid has anchor $\Pi^\sharp:T^*M\to TM$. Integrability means that this algebroid admits a Lie-groupoid integration in the intended smooth category.

The source refers to the canonical symplectic integration, conventionally the source-simply-connected symplectic groupoid $\mathcal G\rightrightarrows M$ with multiplicative symplectic form $\Omega$. Multiplicativity means \[ m^*\Omega=\operatorname{pr}_1^*\Omega+\operatorname{pr}_2^*\Omega \] on composable pairs. The exact convention, including possible non-Hausdorff groupoids, must be checked in the report.

The stated construction obtains this object by symplectic reduction in an infinite-dimensional Banach manifold. Under what conditions can that infinite-dimensional setting be bypassed, giving an appropriate finite-dimensional construction of the intended canonical integration?

The desired distinction concerns the construction, not the dimension of the final Lie groupoid: the usual symplectic integration is itself finite-dimensional, of dimension $2\dim M$. The question is therefore not whether integrability already yields some finite-dimensional output. It seeks conditions enabling a method that avoids the path-space Banach reduction while still producing the required global integration.

The source does not specify an allowed class of finite-dimensional methods, a reduction space or action, finiteness of steps, extra hypotheses on $M$, or an exact necessity/sufficiency criterion. A local symplectic groupoid, an integration with a different source-fiber topology, and the canonical global integration are different outputs and cannot be interchanged silently. No compactness, properness, regularity of the Poisson rank, or Hausdorffness is added as an unstated hypothesis.

Short English statement

The question asks when the canonical global symplectic integration can be constructed without infinite-dimensional path-space reduction. The final groupoid is already finite-dimensional; the missing target is the kind of alternative construction and the conditions that permit it.

Sources

• Local UnsolvedMath registry, permanent ID 30000761, code catalogue entry, "Finite-Dimensional Integration of Poisson Manifolds." Proposed year 2007; proposer unspecified.
• Poisson Geometry and Applications (2008), Oberwolfach Reports, pp. 1243--1298, DOI 10.4171/owr/2007/23; https://doi.org/10.4171/owr/2007/23. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "Birtea--Ortega--Ratiu, related finite-dimensional convexity framework": https://doi.org/10.1090/S0002-9947-08-04689-8. The record gives no theorem connecting that framework to the requested general integration method.
• Statement status corrected_verified; the supplied note says official-report content was checked and wording/notation normalized without changing the claim.
• Literature triage checked 2026-08-21: open. It reports standard constructions and special cases but no general replacement for infinite-dimensional path-space integration. Exact construction scope and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.