The quantization of the symplectic groupoid of the standard Podle s ̀ sphere

Abstract

We give an explicit form of the symplectic groupoid G(S 2 , π) that integrates the semiclassical standard Podles sphere (S 2 , π). We show that Sheu's groupoid G S , 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.

Quantization of the symplectic groupoid

“…We claim that this groupoid appears as groupoid of Bohr-Sommerfeld leaves of some multiplicative integrable model on the symplectic groupoid of the underlying Poisson manifold. This project started from the simplest Poisson homogeneous space of Poisson Lie groups, the Podles sphere in [2], and will be continued in [3]. In this short note we present the case of the Podles sphere studied in [2] from the point of view of integrable models.…”

“…Let us describe very briefly how its symplectic groupoid is defined. Details can be found in [2]. As a manifold, the symplectic groupoid G (S 2 , π) is T * S 2 .…”

“…In [2] it is introduced also a complex polarization so that the general construction of geometric quantization can be applied. The Hilbert algebra described in (5.2) can be recovered from the definition of a convolution product on the space of polarized sections.…”

The modular class as a quantization invariant: a study case

“…Together with a (left) Haar system this defines a measure ν µ c on the whole G bs F (which turns (G bs F , ν µ c ) into a measured groupoid), a KMS state φ µ c and a left Hilbert algebra structure on L 2 (G bs F , ν −1 µ c ) with a modular operator. In [2] we started the program of understanding this quantization procedure for Poisson Lie groups and their homogeneous spaces, in particular we want to recover the results of A.J.L. Sheu characterizing the C * -algebras of the quantum spaces as groupoid C * -algebras [10].…”

“…In this note we will discuss in detail the case of a specific Poisson structure on the 2-sphere (to be called a θ -sphere) determined by an R 2 -action. Our aim is not to produce yet another example of the general program, but, rather, to compare the outcome with the analogous quantization of the Podleś sphere carried through in [2]. These two Poisson structures on S 2 share the same symplectic foliation (one zero point and symplectic complement) and differ by the degree of zero of the bivector at the singular point.…”