Quantization of the symplectic groupoid

Abstract

We present the geometric quantization of the standard Podles sphere by using a multiplicative real polarization of the symplectic groupoid. We introduce the concept of multiplicative integrability of the modular function as one key point of the construction.

The modular class as a quantization invariant: a study case

“…As mentioned in the introduction we will look for a possibly singular multiplicative lagrangian distribution such that the modular 1-cocycle descends to the leaf groupoid. It is natural to seek for a modular multiplicative integrable system, as we considered in [3], i.e. a maximal set F of functions in involution, almost everywhere independent, generating the modular cocycle c V (3.5) and such that the space of level sets G F (S 2 θ ) inherits the groupoid structure.…”