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 on a symplectic manifold 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].
@article{bwmeta1.element.bwnjournal-article-bcpv40z1p329bwm,
author = {Omori, Hideki and Miyazaki, Naoya and Yoshioka, Akira and Maeda, Yoshiaki},
title = {Noncommutative 3-sphere as an example of noncommutative contact algebras},
journal = {Banach Center Publications},
volume = {38},
year = {1997},
pages = {329-334},
zbl = {0890.17022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv40z1p329bwm}
}
Omori, Hideki; Miyazaki, Naoya; Yoshioka, Akira; Maeda, Yoshiaki. Noncommutative 3-sphere as an example of noncommutative contact algebras. Banach Center Publications, Tome 38 (1997) pp. 329-334. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv40z1p329bwm/
[000] [1] F. Bayan, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, Ann. of Physics 111 (1978), 61-110. | Zbl 0377.53024
[001] [2] F. Berezin, General concept of quantization, Comm. Math. Phys. 40 (1975), 153-174. | Zbl 1272.53082
[002] [3] M. Cahen, S. Gutt and J. Rawnsley, Quantization of Kähler manifolds, II, Trans. Amer. Math. Soc. 337 (1993), 73-98. | Zbl 0788.53062
[003] [4] A. V. Karabegov, Deformation quantization with separation of variables on a Kähler manifolds, to appear. | Zbl 06570885
[004] [5] V. Guillemin, Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys. 35 (1995), 85-89. | Zbl 0842.58041
[005] [6] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Noncommutative 3-sphere: A model of noncommutative contact algebras, to appear. | Zbl 1024.53058
[006] [7] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Poincaré-Cartan class and deformation quantization, to appear. | Zbl 0926.53035