A Groupoid Approach to Quantization

@article{Hawkins2006AGA,
  title={A Groupoid Approach to Quantization},
  author={Eli Hawkins},
  journal={Journal of Symplectic Geometry},
  year={2006},
  volume={6},
  pages={61-125}
}
  • E. Hawkins
  • Published 13 December 2006
  • Mathematics
  • Journal of Symplectic Geometry
Many interesting $C∗$-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution $C∗$-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, including geometric quantization of symplectic manifolds and the $C∗$-algebra of a Lie groupoid. I… 
Geometric Quantization of Poisson Manifolds via Symplectic Groupoids
The theory of Lie algebroids and Lie groupoids is a convenient framework for studying properties of Poisson Manifolds. In this work we approach the problem of geometric quantization of Poisson
Quantization of Poisson Manifolds from the Integrability of the Modular Function
We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, combining the tools of geometric quantization with the results of Renault’s theory of
Groupoids, Loop Spaces and Quantization of 2-PLECTIC Manifolds
We describe the quantization of 2-plectic manifolds as they arise in the context of the quantum geometry of M-branes and non-geometric flux compactifications of closed string theory. We review the
Geometric quantization of symplectic and Poisson manifolds
The first part of this thesis provides an introduction to recent development in geometric quantization of symplectic and Poisson manifolds, including modern refinements involving Lie groupoid theory
Symplectic groupoids for cluster manifolds
Quantization of Planck's Constant
This paper is about the role of Planck's constant, $\hbar$, in the geometric quantization of Poisson manifolds using symplectic groupoids. In order to construct a strict deformation quantization of a
Quantization of 2-Plectic Manifolds
We describe an extension of the axioms of quantization to the case of 2-plectic manifolds. We show how such quantum spaces can be obtained as stable classical solutions in a zero-dimensional
Relational symplectic groupoids and Poisson sigma models with boundary
We introduce the notion of relational symplectic groupoid as a way to integrate Poisson manifolds in general, following the construction through the Poisson sigma model (PSM) given by Cattaneo and
Quantization of 2-Plectic Manifolds 1
manifolds. We show how such quantum spaces can be obtained as stable classical solutions in a zero-dimensional 3-algebra reduced model obtained by dimensional reduction of the
Quantization on manifolds with an embedded submanifold
We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study
...
...

References

SHOWING 1-10 OF 67 REFERENCES
Deformation quantization of Heisenberg manifolds
ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms
Extensions of symplectic groupoids and quantization.
An important role of Poisson manifolds is äs intermediate objects between ordinary manifolds, with their commutative algebras of functions, and the "noncommutative spaces" of quantum mechanics. Up to
Quantization of Multiply Connected Manifolds
The standard (Berezin-Toeplitz) geometric quantization of a compact Kähler manifold is restricted by integrality conditions. These restrictions can be circumvented by passing to the universal
Quantization of Poisson algebras associated to Lie algebroids
We prove the existence of a strict deformation quantization for the canonical Poisson structure on the dual of an integrable Lie algebroid. It follows that any Lie groupoid C*-algebra may be regarded
Lie Group Convolution Algebras as Deformation Quantizations of Linear Poisson Structures
Introduction. Let L be a finite dimensional Lie algebra over the real numbers, R, and let L* be its dual vector space. It is well-known [24] that the Lie algebra structure on L defines a natural
Pseudodifferential operators on differential groupoids
We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction
Deformation quantization of pseudo-symplectic (Poisson) groupoids
Abstract.We introduce a new kind of groupoid—a pseudo-étale groupoid, which provides many interesting examples of noncommutative Poisson algebras as defined by Block, Getzler, and Xu. Following the
Fell bundles over groupoids
We study the C*-algebras associated to Fell bundles over groupoids and give a notion of equivalence for Fell bundles which guarantees that the associated C*-algebras are strong Morita equivalent. As
Geometric Models for Noncommutative Algebras
UNIVERSAL ENVELOPING ALGEBRAS Algebraic constructions The Poincare-Birkhoff-Witt theorem POISSON GEOMETRY Poisson structures Normal forms Local Poisson geometry POISSON CATEGORY Poisson maps
On Primitive Ideal Spaces of C*-Algebras over Certain Locally Compact Groupoids
Let F be a locally compact Hausdorff second countable groupoid with a left Haar system {v x } x∈X in the sense of [9] (X = the unit space of Γ). By analogy with Fell’s algebraic bundles over groups,
...
...