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

General

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 symplectic groupoids are used to discuss geometric quantization not just of symplectic manifolds but more generally of Poisson manifolds.

See geometric quantization of symplectic groupoids.

As Lie integration of Poisson Lie algebroid

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Related concepts

• symplectic manifold

• symplectic Lie n-algebroid

• symplectic infinity-groupoid

References

The notion of symplectic groupoids was apparently proposed independently by Karasëv, Weinstein, and Zakrzewski, all motiviated from the problem of quantization.

• Alan Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras in Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), 281–290, Springer, New York, (1991) MR1104934.

• Alan Weinstein, Tangential deformation quantization and polarized symplectic groupoids, in Deformation theory and symplectic geometry (Ascona, 1996), 301–314, Kluwer (1997) MR1480730

The formal groupoid version of symplectic groupoids is discussed in

• Alberto Cattaneo, Benoit Dherin, Giovanni Felder, Formal symplectic groupoid (arXiv:0312380)

For more references see geometric quantization of symplectic groupoids .