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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.

MSC:
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
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