zbMATH — the first resource for mathematics

Quantization of quadratic Poisson brackets on a polynomial algebra of three variables. (English) Zbl 0934.16025
The authors list all possible quadratic Poisson brackets on a polynomial algebra in three variables and show explicit quantizations of all of them; relations with Sklyanin algebras are pointed out. Since the submission of this paper (April 1995), Kontsevich gave an explicit method to quantize Poisson brackets on differentiable manifolds, and in particular on \(\mathbb{R}^n\) [see M. Kontsevich, q-alg/9709040]. Reshetikhin announced recently explicit quantization for all quadratic Poisson brackets on polynomial algebras, using Kontsevich’s method.

16S80 Deformations of associative rings
16W50 Graded rings and modules (associative rings and algebras)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B63 Poisson algebras
Full Text: DOI arXiv
[1] Artin, M.; Schelter, W., Graded algebras of global dimension 3, Adv. math., 66, 171-216, (1987) · Zbl 0633.16001
[2] Artin, M.; Tate, J.; Van Den Bergh, M., Some algebras associated to automorphisms of elliptic curves, () · Zbl 0744.14024
[3] Bakelin, J., A distributiveness property of augmented algebras and some related homological results, (1981), Univ. Stockholm, Preprint
[4] Bergman, G.M., The diamond lemma for ring theory, Adv. math., 29, 178-218, (1978) · Zbl 0326.16019
[5] Bourbaki, N., Algèbre commutative, (1961), Hermann Paris · Zbl 0108.04002
[6] Cohn, P.M., Skew field constructions, (1977), Cambridge University Press Cambridge · Zbl 0355.16009
[7] Drinfeld, V., On quadratic commutator relations in the quasiclassical case, ()
[8] Dufour, J.; Haraki, A., Rotationnels et structures de Poisson quadratiques, C.R. acad. sci. Paris, 312, I, 137-140, (1991) · Zbl 0719.58001
[9] Feigin, B.; Odessky, A., Sklyanin algebras associated with elliptic curves, (1989), Institute for Theoretical Physics Ukraine, Kiev, Preprint
[10] Feigin, B.; Odessky, A., Elliptic Sklyanin algebras, Functional anal. appl., 23, 3, (1989)
[11] Liu, Z.; Xu, P., On quadratic Poisson structures, Lett. math. phys., 26, 33-42, (1992) · Zbl 0773.58007
[12] Odessky, A., Infinite dimensional algebras and complex varieties, ()
[13] Omori, H.; Maedand, Y.; Yoshioka, A., Deformation quantizations of Poisson algebras, Contemporary math., 179, 229-242, (1994)
[14] Polistchuk, A.; Posicelsky, L., On quadratic algebras, (1991), Preprint, Moscow
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.