or dictionary between the world of symplectic manifold (classical level) and the world of linear spaces(quantum level). This correspondence, as explained in [@weinstein1997], is summarized in the following table:

   Symplectic world         Linear world
  ------------------ ---------------------------
         $M$                   $Q(M)$
    $L\subset M $          $Q(L) \in Q(M)$
    $\overline{M}$    $Q(\overline{M})= Q(M)^*$
    $Q(M\times N)$       $Q(M)\otimes Q(N)$

Here $M$ is a symplectic manifold, $\overline{M}$ the same manifold with opposite symplectic structure, $L$ a Lagrangian submanifold, and $Q(M)$ a complex vector space. $Q$ stands for the “Quantization functor". In particular, canonical relations, i.e., Lagrangian submanifolds of $\overline M\times N$ are sent by $Q$ to linear maps from $Q(M)$ to $Q(N)$. The main ingredient is the assumption that quantization is functorial , i.e., the composition of canonical relations should be sent to the composition of linear maps (see [@weinstein1991]). If such a quantization functor existed, we could ask the following question:

*To what kind of symplectic manifold should we associate an algebra( i.e., a vector space with an associative product)?*

Answering this question leads directly to the notion of symplectic groupoid, see [@weinstein1996].

A symplectic groupoid is a Lie groupoid $G$ (see [@weinstein1997] for a precise definition of a Lie groupoid) with a symplectic form $\omega$ for which the 