## On the quantization of Poisson brackets.(English)Zbl 0931.53040

In this paper, which appeared considerably earlier than the famous preprint of M. Kontsevich “Formal quantization of Poisson manifolds”, the author introduces a rather wide class of Poisson brackets which can be quantized by an algebraic version of B. V. Fedosov’s method [Funct. Anal. Appl. 25, 184-194 (1991; Zbl 0737.47042)]. An interesting, purely algebraic consequence of these results is that any Poisson bracket on a field of zero characteristic can be quantized.

### MSC:

 53D17 Poisson manifolds; Poisson groupoids and algebroids 53D55 Deformation quantization, star products

Zbl 0737.47042
Full Text:

### References:

 [1] Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D., Deformation theory and quantization, Ann. physics, 111, 61-110, (1978) · Zbl 0377.53024 [2] Gerstenhaber, M., On the deformation of rings and algebras, Ann. of math., 79, 59-103, (1964) · Zbl 0123.03101 [3] Gerstenhaber, M., On the deformation of rings and algebras, II, Ann. of math., 84, 1-19, (1966) · Zbl 0147.28903 [4] Gerstenhaber, M., On the deformation of rings and algebras, III, Ann. of math., 88, 1-34, (1968) · Zbl 0182.05902 [5] Gerstenhaber, M., On the deformation of rings and algebras, IV, Ann. of math., 99, 257-276, (1974) · Zbl 0281.16016 [6] M. Gerstenhaber, S. D. Schack, Algebraic cohomology and deformation theory, Deformation Theory of Algebra and Structures and Applications, NATO-ASI Series 247 · Zbl 0676.16022 [7] V. G. Drinfeld, Quantum groups, Proceedings, Internat. Congr. Mathematicians, Berkley, CA 1986, 1, Academic Press [8] J. Donin, L. Makar-Limanov, On quadratic Poisson brackets on a polynomial ring of three variables [9] Donin, J.; Gurevich, D.; Majid, S., R, Ann. inst. H. Poincaré, 58, 235-246, (1993) · Zbl 0783.17005 [10] Donin, J.; Gurevich, D., Quasi-Hopf algebras andR, Selecta math. soviet., 12, 37-48, (1993) · Zbl 0811.17017 [11] Donin, J.; Gurevich, D., Some Poisson structures associated to Drinfeld-JimboR, Israel math. J., 92, 23-32, (1995) · Zbl 0842.58039 [12] Donin, J.; Shnider, S., Quantum symmetric spaces, J. pure appl. algebra, 100, 103-115, (1995) · Zbl 0838.17019 [13] Illusie, L., Complexe cotangent et déformations I, Lecture notes in math., 239, (1971) [14] M. de Wilde, P. Lecomte, Formal deformations of the Poisson Lie algebra of a symplectic manifold and star-products, Deformation Theory of Algebra and Structures and Applications, NATO-ASI Series 247, Kluwer Academic, Dordrecht [15] D. Mélotte, Invariant deformations of the Poisson Lie algebras of a symplectic manifold and star-products, Deformation Theory of Algebra and Structures and Applications, NATO-ASI Series 247, Kluwer Academic, Dordrecht [16] Fedosov, B.V., Theorems on index, Itogi mauki i tekhniki, Viniti, 65, (1991) · Zbl 0884.58087 [17] Fedosov, B.V., Deformation quantization and asymptotic operator representation, Funct. anal. appl., 25, 165-268, (1991) · Zbl 0737.47042 [18] Omori, H.; Maeda, Y.; Yoshioka, A., Weyl manifolds and deformation quantization, Adv. math., 85, 224-255, (1991) · Zbl 0734.58011 [19] Grauert, H.; Remmert, R., Coherent analytic sheaves, (1984), Springer-Verlag New York/Berlin
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.