Quantization of quadratic Poisson brackets on a polynomial algebra of three variables

Abstract

Poisson brackets (P.b) are the natural initial terms for the deformation quantization of commutative algebras. There is an open problem whether any Poisson bracket on the polynomial algebra of $n$ variables can be quantized. It is known (Poincare-Birkhoff-Witt theorem) that any linear 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 unitial terms for the quantization of polynomial algebras 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 quadratic P.b can be quantized. Moreover, the quantization is given as the quotient algebra of tensor algebra of three 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 variables, which we also give in the paper. In Appendix we give explicit algebraic constructions of the quantized algebras appeared here to show that they are related to algebras of global dimension three considered by M.Artin, W.Schelter, J.Tate, and M.Van Den Bergh from a different point of view.