Reference : Quantization of Poisson manifolds from the integrability of the modular function
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/13522
Quantization of Poisson manifolds from the integrability of the modular function
English
Bonechi, Francesco [FLorence University]
Tarlini, Marco [Florence University]
Ciccoli, Nicola [University of Perugia]
Qiu, Jian mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
2014
Communications in Mathematical Physics
Springer Science & Business Media B.V.
331
2
851–885
Yes (verified by ORBilu)
International
0010-3616
[en] Quantization ; Poisson-Lie group ; symplectic groupoid
[en] We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows very singular polarizations. In particular we consider the case when the modular function is "multiplicatively integrable", i.e. when the space of leaves of the polarization inherits a groupoid structure. If suitable regularity conditions are satisfied, then one can define the quantum algebra as the convolution algebra of the subgroupoid of leaves satisfying the Bohr-Sommerfeld conditions. We apply this procedure to the case of a family of Poisson structures on CP_n, seen as Poisson homogeneous spaces of the standard Poisson-Lie group SU(n+1). We show that a bihamiltoniam system on CP_n defines a multiplicative integrable model on the symplectic groupoid; we compute the Bohr-Sommerfeld groupoid and show that it satisfies the needed properties for applying Renault theory. We recover and extend Sheu's description of quantum homogeneous spaces as groupoid C*-algebras.
Researchers
http://hdl.handle.net/10993/13522
10.1007/s00220-014-2050-9
http://arxiv.org/abs/1306.4175

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