QUANTIZATION OF QUADRATIC POISSON BRACKETS ON A POLYNOMIAL ALGEBRA OF3 VARIABLES

Poisson brackets (P.b.) are the natural initial terms for the deformat ion quantization of commutative algebras. There is an open problem whe ther any Poisson bracket on the polynomial algebra of n variables can be quantized. It is known (Poincare-Birkhoff-Witt theorem) that any li near P.b. for all n can be quantized. On the other hand, it is easy to show that in case n=2 any P.b. is quantizable as well. Quadratic P.b. appear as the initial terms for the quantization of polynomial algebr as as quadratic algebras. The problem of the quantization of quadratic P.b. is also open. In the paper we show that in case n=3 any quadrati c P.b, can be quantized. Moreover, the quantization is given as the qu otient algebra of tensor algebra of tk-ee variables by relations which are similar to those in the Poincare-Birkhoff-Witt theorem. The proof uses a classification of all quadratic Poisson brackets of three vari ables, which vie also give in the paper. Ln the appendix we give expli cit algebraic constructions of the quantized algebras appeared here an d show that they are related to algebras of global dimension three con sidered by M. Artin, W. Schelter, J. Tate, M. Van Den Bergh and other authors from a different point of view. (C) 1998 Elsevier Science B.V. All rights reserved.