Noncommutative 3-sphere as an example of noncommutative contact algebras

• Autorzy: Hideki Omori , Naoya Miyazaki , Akira Yoshioka , Yoshiaki Maeda
• Języki publikacji: EN

Abstrakty

EN

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].

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1997
• Tom: 40
• Numer: 1
• Strony: 329-334

Daty

wydano

1997

Twórcy

autor

Hideki Omori

• Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba, 278, Japan

autor

Naoya Miyazaki

• Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba, 278, Japan

autor

Akira Yoshioka

• Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo, Noda, Chiba, 278, Japan

autor

Yoshiaki Maeda

• Department of Mathematics Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama, 223, Japan

Bibliografia

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