The quantization of the symplectic groupoid of the standard Podlès sphere (Q432282)

The simplest example of quantum homogeneous space, namely the standard Podleś sphere \((\mathbb{S}^2, \pi )\), is analyzed in this paper from the point of view of geometric quantization; here \(\pi \) is the Poisson structure. More precisely, the symplectic groupoid of \((\mathbb{S}^2, \pi )\) is completely described as well as a partition of this symplectic groupoid in Lagrangian submanifolds whose Bohr-Sommerfeld leaves reproduce the Sheu groupoid. Two distinct polarizations of the symplectic groupoid are obtained; the first one is real and singular, the second being complex and recovers the convolution algebra on the space of polarized sections.