On the integration of Poisson homogeneous spaces

Bonechi, Francesco; Ciccoli, Nicola; Staffolani, Nicola; Tarlini, M.

doi:10.1016/j.geomphys.2008.07.001

Cited by 10 publications

(18 citation statements)

References 15 publications

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“…Since (S 2 , π) is a Poisson homogeneous space, i.e. S 2 = U(1)\SU (2), with SU(2) equipped with the Poisson Lie group structure, we can use the general construction given in [1]. The general framework for Poisson homogeneous spaces is the following, see [14] for a detailed account of Poisson reduction.…”

Section: The Podles Spheresmentioning

confidence: 99%

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

Bonechi¹,

Ciccoli²,

Staffolani³

et al. 2012

Journal of Geometry and Physics

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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.

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Section: The Podles Spheresmentioning

confidence: 99%

“…The general result established in [1] describes a symplectic groupoid of the Poisson homogeneous space (H\G, π H\G ) as…”

Section: The Podles Spheresmentioning

confidence: 99%

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

Bonechi¹,

Ciccoli²,

Staffolani³

et al. 2012

Journal of Geometry and Physics

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“…Although the work in [9] involves realizing C CP n q,c as part of a concrete groupoid C * -algebra in order to analyze the algebra structure and extract useful information, it is not clear whether one can actually realize C CP n q,c as a groupoid C * -algebra itself. However from a purely differential geometric consideration, an elegant program of constructing some quantum homogeneous spaces as the groupoid C * -algebras of geometrically constructed Bohr-Sommerfeld groupoids is later successfully developed by Bonechi, Ciccoli, Qiu, Staffolani, Tarlini [1,2]. Naturally, it is of great interest to decide whether the quantum complex projective space arising from this new program is indeed the same as the known version of C CP n q,c .…”

Section: Introductionmentioning

confidence: 99%

Nonstandard Quantum Complex Projective Lin

Ciccoli

Sheu

2020

SIGMA

Self Cite

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In our attempt to explore how the quantum nonstandard complex projective spaces CP n q,c studied by Korogodsky, Vaksman, Dijkhuizen, and Noumi are related to those arising from the geometrically constructed Bohr-Sommerfeld groupoids by Bonechi, Ciccoli, Qiu, Staffolani, and Tarlini, we were led to establish the known identification of C CP 1 q,c with the pull-back of two copies of the Toeplitz C *-algebra along the symbol map in a more direct way via an operator theoretic analysis, which also provides some interesting nonobvious details, such as a prominent generator of C CP 1 q,c being a concrete weighted double shift.

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“…Different methods have been employed in the literature to tackle integrability of Poisson homogeneous spaces, producing partial results e.g. in [6,28,43,44,56,58] (see also [7,8] for applications to quantization). In this paper, we prove integrability of Poisson homogeneous spaces in its full generality.…”

Section: Introductionmentioning

confidence: 99%

Dirac geometry and integration of Poisson homogeneous spaces

Bursztyn¹,

Iglesias-Ponte²,

Lu³

2019

Preprint

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Using tools from Dirac geometry, we show through an explicit construction that any Poisson homogeneous space of any Poisson Lie group admits an integration to a symplectic groupoid. Our theorem follows from a more general result which relates, for a principal bundle M → B, integrations of a Poisson structure on B to integrations of its pullback Dirac structure on M by pre-symplectic groupoids. Our construction gives a distinguished class of explicit real or holomorphic pre-symplectic and symplectic groupoids over semi-simple Lie groups and some of their homogeneous spaces, including their symmetric spaces, conjugacy classes, and flag varieties. Contents 1. Introduction 2. Preliminaries on Dirac geometry 2.1. Dirac structures and pre-symplectic groupoids 2.2. Invariant Dirac structures and Poisson quotients 2.3. Integrability of Poisson structures on quotients 3. Integration of homogeneous Dirac structures on Poisson Lie groups 3.1. Review of definitions 3.2. Explicit integrations 4. Integration of Poisson homogeneous spaces 4.1. Explicit integrations 4.2. Proof of the main theorem 5. The holomorphic setting 6. Examples: old and new 6.1. Special cases and previous examples 6.2. Examples related to semi-simple Lie groups Appendix A. The pre-symplectic groupoid (G(L), ω) A.1. Deriving the formula for ω A.2. Proof of Theorem 3.12

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On the integration of Poisson homogeneous spaces

Cited by 10 publications

References 15 publications

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

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

Nonstandard Quantum Complex Projective Lin

Dirac geometry and integration of Poisson homogeneous spaces

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