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

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-BirkhoffWitt 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. 1 Deformations of quadratic algebras Let A be an associative algebra with unit over a field k of characteristic zero. We will consider deformations of A over the algebra of formal power series k[[h̄]] in a variable h̄. By a deformation of A we mean an algebra Ah̄ over k[[h̄]] which is isomorphic to A[[h̄]] = A⊗̂kk[[h̄]] as a k[[h̄]]-module and Ah̄/h̄Ah̄ = A (the symbol ⊗̂ denotes the tensor product completed in the h̄-adic topology). We will also denote A as A0. If A′h̄ is another deformation of A, we call the deformations Ah̄ and A ′ h̄ equivalent if there exists a k[[h̄]]-algebra isomorphism Ah̄ → A ′ h̄ which induces the identity automorphism of A0. Let T = T (V ) be a tensor algebra over a finite-dimensional vector space V over a field k and let A be a quotient algebra of T , i.e., A = T/I where I is an ideal in T . It is easy to see that if Ah̄ is a deformation of A, then Ah̄ = T [[h̄]]/Ih̄ where Ih̄ is an ideal in T [[h̄]] such that I = Ih̄/h̄Ih̄. Consider T as a graded algebra, T = ⊕kT , and let A also be a graded algebra, i.e., I is a graded ideal. Denote by I the k-th homogeneous component of I, I = I ∩ T . If Ah̄ is