The quantization of the symplectic groupoid of the standard Podles sphere

Abstract

We give an explicit form of the symplectic groupoid that integrates the semiclassical standard Podles sphere. We show that Sheu's groupoid, whose convolution C*-algebra quantizes the sphere, appears as the groupoid of the Bohr-Sommerfeld leaves of a (singular) real polarization of the symplectic groupoid. 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.

33 pages; minor corrections

References in corpus (3)

• On the integration of Poisson homogeneous spaces
• Integrability and reduction of Poisson group actions
• On the Hochschild (co)homology of Quantum Homogeneous Spaces

Cited by in corpus (7)

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• Groupoid Quantization of Loop Spaces
• Quantization of Poisson manifolds from the integrability of the modular function
• Dirac geometry and integration of Poisson homogeneous spaces
• Mutation Symmetries in BPS Quiver Theories: Building the BPS Spectra
• Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
• Symplectic groupoids for cluster manifolds