nLab
symplectic groupoid

Idea

A Poisson manifold may be thought of as a Poisson Lie algebroid, a Lie algebroid with extra structure: called an n-symplectic manifold for $n=1$.

By Lie integration this Lie algebroid should integrate to a Lie groupoid with extra structure. Symplectic groupoids are supposed to be these objects that integrate n-symplectic manifold aka Poisson manifolds in this sense.

The groupoid algebra of these symplectic groupoids are C-star algebras that may be regarded as the quantization of the original Poisson manifold. This is described in the references below.

Definition

A symplectic Lie groupoid is a Lie groupoid $C$ whose space of objects is a Poisson manifold and whose space of morphisms carries a symplectic structure whose symplectic form $\omega \in {\Omega }_{\mathrm{closed}}^2(\mathrm{Mor}(C))$ is multiplicative in that it is closed regarded as an element in the simplicial deRham complex of the nerve of $C$:

$$0=d\omega +\delta \omega =\delta \omega ={\mathrm{pr}}_1^{*}\omega -{\mathrm{compose}}^{*}\omega +{\mathrm{pr}}_2^{*}\omega$$0 = d \omega + \delta \omega = \delta \omega = pr_1^* \omega - compose^* \omega + pr_2^* \omega

Properties

Every Lie groupoid integrating a Poisson Lie algebroid is naturally a symplectic Lie groupoid. Picking always the unique source-simply connected integrating Lie groupoid produces a functor

$$\Sigma :\mathrm{PoissonManifolds}\to \mathrm{SymplecticGroupoids}.$$\Sigma : PoissonManifolds \to SymplecticGroupoids \,.

When the Poisson manifold we start with happens to be a symplectic manifold, then its symplectic Lie groupoid is always the fundamental groupoid of $X$:

$$((X,\pi )\mathrm{symplectic})\Rightarrow (\Sigma (X,\pi )=\Pi (X)).$$((X,\pi) symplectic) \;\;\Rightarrow\;\; (\Sigma(X,\pi) = \Pi(X)) \,.

When $X$ is simply connected such that $\Pi (X)$ is the codiscrete groupoid $\mathrm{Pair}(X)$ we have that the symplectic form on $\mathrm{Mor}(\Pi (X))=X\times X$ is $\omega \otimes (-\omega )$, for $\omega$ the symplectic form on $X$.

in geometric quantization of Poisson manifolds

In the groupoid approach to quantization

• Eli Hawkins, A groupoid approach to quantization (arXiv)

symplectic groupoids are the right tool for thinking about geometric quantization not of symplectic manifolds but of just Poisson manifolds.

A symplectic groupoid in this context is something like a groupoid internal to the category of symplectic manifolds

but beware, there is some fine print, see the references below

The following blog discussion should eventually be copied here:

• Urs Schreiber,

  • A Groupoid approach to quantization

  • Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids

  • Eli Hawkins on Geometric Quantization I

  • Eli Hawkins on Geometric Quantization II

References

• Eli Hawkins, A groupoid approach to quantization (arXiv)