The quantization of the symplectic groupoid of the standard Podles ̀ sphere

We give an explicit form of the symplectic groupoid G(S^2,π) that integrates the semiclassical standard Podles̀ sphere (S^2,π). We show that Sheu’s groupoid GS, whose convolution C∗-algebra quantizes the sphere, appears as the groupoid of the Bohr–Sommerfeld leaves of a (singular) real polarization of G(S^2,π). By using a complex polarization we recover the convolution algebra on the space of polarized sections. We stress the role of the modular class in the definition of the scalar product in order to get the correct quantum space.