multiplication space $G^{(m)} =\{(x,y,x\bullet y)/x,y\in G \textrm{ are composable elements}\}$ is a Lagrangian submanifold of $\overline G\times\overline G\times G$ ($\overline G$ being the symplectic manifold with symplectic form $-\omega$). It can be shown (see [@weinstein1987]) that, given a symplectic groupoid $G$, there is an induced Poisson structure on the base space $G^{(0)}$. Conversely, given a Poisson manifold $P$ we call symplectic groupoid over $P$ any symplectic groupoid $G$ such that the base space $G^{(0)}$ is diffeomorphic as Poisson manifold to $P$. In this case we say that $G$ integrates $P$ and we call integrable Poisson manifolds the Poisson manifolds for which we can find such a $G$.

Applying the “Quantization functor" $Q$ to the symplectic groupoid $G$, we should then get a vector space $Q(G)$ and an associative product $Q(G^{(m)})$ on it. The associativity of this product being guaranteed by the associativity of the groupoid multiplication and the functoriality of $Q$.

These facts suggest the following procedure to quantize Poisson manifolds $P$:

1.  Find a symplectic groupoid $G$ such that the base $G^{(0)}$ is diffeomorphic to the Poisson manifold $P$ .

2.  Quantize (geometric quantization,...) $G$ and $G^{(m)}$ to get the quantum algebra.

This is the idea of quantization by 