Noncommutative 3-sphere as an example of noncommutative contact algebras

Hideki Omori; Naoya Miyazaki; Akira Yoshioka; Yoshiaki Maeda

Banach Center Publications (1997)

  • Volume: 40, Issue: 1, page 329-334
  • ISSN: 0137-6934

Abstract

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The notion of deformation quantization was introduced by F.Bayen, M.Flato et al. in [1]. The basic idea is to formally deform the pointwise commutative multiplication in the space of smooth functions C ∞ ( M ) on a symplectic manifold M to a noncommutative associative multiplication, whose first order commutator is proportional to the Poisson bracket. It is of interest to compute this quantization for naturally occuring cases. In this paper, we discuss deformations of contact algebras and give a definition of deformations of algebras slightly different from the deformation quantization of Poisson algebras. Since the standard 3-sphere is a basic example of a contact manifold, we study the properties of the noncommutative 3-sphere obtained by this reduction. We remark that the parameter of the deformation of a contact algebra is not in the center, while the deformation quantization of Poisson algebras is given by algebras of formal power series of functions on a manifold; in particular, the deformation parameter is a central element. Details and related results will appear in [6] and [7].

How to cite

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Omori, Hideki, et al. "Noncommutative 3-sphere as an example of noncommutative contact algebras." Banach Center Publications 40.1 (1997): 329-334. <http://eudml.org/doc/252185>.

@article{Omori1997,
abstract = {The notion of deformation quantization was introduced by F.Bayen, M.Flato et al. in [1]. The basic idea is to formally deform the pointwise commutative multiplication in the space of smooth functions $C^∞(M)$ on a symplectic manifold $M$ to a noncommutative associative multiplication, whose first order commutator is proportional to the Poisson bracket. It is of interest to compute this quantization for naturally occuring cases. In this paper, we discuss deformations of contact algebras and give a definition of deformations of algebras slightly different from the deformation quantization of Poisson algebras. Since the standard 3-sphere is a basic example of a contact manifold, we study the properties of the noncommutative 3-sphere obtained by this reduction. We remark that the parameter of the deformation of a contact algebra is not in the center, while the deformation quantization of Poisson algebras is given by algebras of formal power series of functions on a manifold; in particular, the deformation parameter is a central element. Details and related results will appear in [6] and [7].},
author = {Omori, Hideki, Miyazaki, Naoya, Yoshioka, Akira, Maeda, Yoshiaki},
journal = {Banach Center Publications},
keywords = {noncommutative geometry; noncommutative contact algebra; regulated algebra; noncommutative three-sphere},
language = {eng},
number = {1},
pages = {329-334},
title = {Noncommutative 3-sphere as an example of noncommutative contact algebras},
url = {http://eudml.org/doc/252185},
volume = {40},
year = {1997},
}

TY - JOUR
AU - Omori, Hideki
AU - Miyazaki, Naoya
AU - Yoshioka, Akira
AU - Maeda, Yoshiaki
TI - Noncommutative 3-sphere as an example of noncommutative contact algebras
JO - Banach Center Publications
PY - 1997
VL - 40
IS - 1
SP - 329
EP - 334
AB - The notion of deformation quantization was introduced by F.Bayen, M.Flato et al. in [1]. The basic idea is to formally deform the pointwise commutative multiplication in the space of smooth functions $C^∞(M)$ on a symplectic manifold $M$ to a noncommutative associative multiplication, whose first order commutator is proportional to the Poisson bracket. It is of interest to compute this quantization for naturally occuring cases. In this paper, we discuss deformations of contact algebras and give a definition of deformations of algebras slightly different from the deformation quantization of Poisson algebras. Since the standard 3-sphere is a basic example of a contact manifold, we study the properties of the noncommutative 3-sphere obtained by this reduction. We remark that the parameter of the deformation of a contact algebra is not in the center, while the deformation quantization of Poisson algebras is given by algebras of formal power series of functions on a manifold; in particular, the deformation parameter is a central element. Details and related results will appear in [6] and [7].
LA - eng
KW - noncommutative geometry; noncommutative contact algebra; regulated algebra; noncommutative three-sphere
UR - http://eudml.org/doc/252185
ER -

References

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  1. [1] F. Bayan, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, Ann. of Physics 111 (1978), 61-110. Zbl0377.53024
  2. [2] F. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174. Zbl1272.53082
  3. [3] M. Cahen, S. Gutt and J. Rawnsley, Quantization of Kähler manifolds, II, Trans. Amer. Math. Soc. 337 (1993), 73-98. Zbl0788.53062
  4. [4] A. V. Karabegov, Deformation quantization with separation of variables on a Kähler manifolds, to appear. Zbl06570885
  5. [5] V. Guillemin, Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys. 35 (1995), 85-89. Zbl0842.58041
  6. [6] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Noncommutative 3-sphere: A model of noncommutative contact algebras, to appear. Zbl1024.53058
  7. [7] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Poincaré-Cartan class and deformation quantization, to appear. Zbl0926.53035

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