Noncommutative 3-sphere A model of noncommutative contact algebras

Hideki OMORI; Yoshiaki MAEDA; Naoya MIYAZAKI; Akira YOSHIOKA

doi:10.2969/jmsj/05040915

Noncommutative 3-sphere

A model of noncommutative contact algebras

Hideki OMORI, Yoshiaki MAEDA, Naoya MIYAZAKI, Akira YOSHIOKA

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JOURNAL FREE ACCESS

1998 Volume 50 Issue 4 Pages 915-943

DOI https://doi.org/10.2969/jmsj/05040915

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Published: 1998 Received: January 09, 1996 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : Noncommutative 3-sphere: A model of noncommutative contact algebras
Right : Noncommutative 3-sphere

Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Masafumi Okumura for his sixtieth birthday
Right : A model of noncommutative contact algebras

Date of correction: October 20, 2006 Reason for correction: - Correction: AFFILIATION Details: Wrong :
1) Department of Mathematics, Faculty of Mathematics, Faculty of Science and Technology, Science University of Tokyo
2) Department of Mathematics, Keino University, Faculty of Science and Technology
3) Department of Mathematics, Faculty of Engineering, Science University of Tokyo

Right :
1) Department of Mathematics, Faculty of Science and Technology, Science University of Tokyo
2) Department of Mathematics, Keio University, Faculty of Science and Technology
3) Department of Mathematics, Faculty of Engineering, Science University of Tokyo

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : [A] A. Weinstein, Deformation quantization, Asterisque 227 (1995), 389-409.
[B] F. Bayan, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, Ann. of Physics 111 (1978), 61-110.
[Be] Berezin, General concept of quantization, Comm. Math. Phys. 8 (1975), 153-174.
[BG] L. Boutet de Monvel, V. Guillemin, The spectral theory of Töplitz operators, vol. Ann. Math. Studies 99, Princeton Univ. Press, 1981.
[CGR] M. Cahen, S. Gutt and J. Rawnsley, Quantization of Kähler manifolds, II, Trans. Amer. Math. Soc. 337 (1993), 73-98.
[CHZ] C. S. Chu, P. M. Ho, B. Zumino, The quantum 2-sphere as a complex quantum manifold, to appear.
[D] P. Dirac, Principles of quantum mechanics, Oxford Press, 1967.
[De] P. Deligne, Déformations de l'algèbre des fonctions d'une variété symplectique: Comparison entre Fedosov et De Wilde, Lecomte., Selecta Math. New Series 1 no. 4 (1995), 667- 397.
[G] V. Guillemin, Star products on compact prequantizable symplectic manifold, Lett. Math. Physics 35 (1995), 85-89.
[GS] V. Guillemin, S. Sternberg, Geometric asymptotics, Amer. Math. Soc, 1977.
[K] A. V. Karabegov, Deformation quantization with separation of variables on a Kähler manifolds, to appear.
[L] A. Lichnerowicz, Les variétés de Jacobi et leure algèbre de Lie associées, J. Math. Pures et Appl. 57 (1978), 453-488.
[M] A. Messiah, Quantum mechanics, I, North-Holland, 1972.
[MO] T. Masuda and H. Omori, The noncommutative algebra of the quantum group SU_q (2) as a quantized Poisson manifold, Contemporary Mathematics 179 (1994), 161-172.
[O] H. Omori, Infinite, dimensional Lie groups, AMS. Translation Monograph, Amer. Math. Soc., 1996.
[OMY1] H. Omori, Y. Maeda and A. Yoshioka, Weyl manifolds and deformation quantization, Adv. in Math. 85 (1991), 224-255.
[OMY2] H. Omori, Y. Maeda and A. Yoshioka, Deformation quantization of Poisson algebras, Contemporary Mathematics 179 (1994), 213-241.
[OMMY1] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Noncommutative contact algebras, Deformation Theory and Symplectic Geometry, eds. D. Sternheimer et al. Kluwer, 1997, 333-338.
[OMMY2] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Poincaré-Cartan class and deformation quantization, Commun. Math. Phys. 194 (1998), 207-230.
[St] P. Štovicek, Quantum line bundles on S² and method of orbits for SU_q(2), J. Math. Phys. 34
[vH] L. Van Hove, Sur certaines représentations unitaires d'un groupe infini de transformations, Mem. de Acad. Roy. de Belgique 26 (1951), 61-102.
[Y] K. Yoshida, Functional Analysis, Springer-Verlag, 1965.

Right : [A] A. Weinstein, Deformation quantization, Asterisque 227 (1995), 389-409.
[B] F. Bayan, M. Flato, C. Fronsdal, A. Lichnerowicz and D. Sternheimer, Deformation theory and quantization, Ann. of Physics 111 (1978), 61-110.
[Be] Berezin, General concept of quantization, Comm. Math. Phys. 8 (1975), 153-174.
[BG] L. Boutet de Monvel, V. Guillemin, The spectral theory of Töplitz operators, vol. Ann. Math. Studies 99, Princeton Univ. Press, 1981.
[CGR] M. Cahen, S. Gutt and J. Rawnsley, Quantization of Kähler manifolds, II, Trans. Amer. Math. Soc. 337 (1993), 73-98.
[CHZ] C. S. Chu, P. M. Ho, B. Zumino, The quantum 2-sphere as a complex quantum manifold, to appear.
[D] P. Dirac, Principles of quantum mechanics, Oxford Press, 1967.
[De] P. Deligne, Déformations de l'algèbre des fonctions d'une variété symplectique: Comparison entre Fedosov et De Wilde, Lecomte., Selecta Math. New Series 1 no. 4 (1995), 667-397.
[G] V. Guillemin, Star products on compact prequantizable symplectic manifold, Lett. Math. Physics 35 (1995), 85-89.
[GS] V. Guillemin, S. Sternberg, Geometric asymptotics, Amer. Math. Soc, 1977.
[K] A. V. Karabegov, Deformation quantization with separation of variables on a Kähler manifolds, to appear.
[L] A. Lichnerowicz, Les variétés de Jacobi et leure algèbre de Lie associées, J. Math. Pures et Appl. 57 (1978), 453-488.
[M] A. Messiah, Quantum mechanics, I, North-Holland, 1972.
[MO] T. Masuda and H. Omori, The noncommutative algebra of the quantum group SU_q (2) as a quantized Poisson manifold, Contemporary Mathematics 179 (1994), 161-172.
[O] H. Omori, Infinite, dimensional Lie groups, AMS. Translation Monograph, Amer. Math. Soc., 1996.
[OMY1] H. Omori, Y. Maeda and A. Yoshioka, Weyl manifolds and deformation quantization, Adv. in Math. 85 (1991), 224-255.
[OMY2] H. Omori, Y. Maeda and A. Yoshioka, Deformation quantization of Poisson algebras, Contemporary Mathematics 179 (1994), 213-241.
[OMMY1] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Noncommutative contact algebras, Deformation Theory and Symplectic Geometry, eds. D. Sternheimer et al. Kluwer, 1997, 333-338.
[OMMY2] H. Omori, Y. Maeda, N. Miyazaki and A. Yoshioka, Poincaré-Cartan class and deformation quantization, Commun. Math. Phys. 194 (1998), 207-230.
[St] P. Štovicek, Quantum line bundles on S² and method of orbits for SU_q (2), J. Math. Phys. 34 (1993), 1606-1613.
[vH] L. Van Hove, Sur certaines représentations unitaires d'un groupe infini de transformations, Mem. de Acad. Roy. de Belgique 26 (1951), 61-102.
[Y] K. Yoshida, Functional Analysis, Springer-Verlag, 1965.

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