ИСТИНА

My talk concerns recent progress made in the positive resolution of Kontsevich's conjecture, which states that, in characteristic zero, deformation quantization of affine space preserves the group of symplectic polynomial automorphisms, i.e. the group of polynomial symplectomorphisms in dimension $2n$ is canonically isomorphic to the group of automorphisms of the corresponding $n$-th Weyl algebra. The conjecture is positive for $n=1$ and open for $n>1$. In the talk, the plan of attack on Kontsevich conjecture called lifting of symplectomorphisms is presented. Starting with a polynomial symplectomorphism, one can lift it to an automorphism of the power series completion of the Weyl algebra (with commutation relations preserved), after which one can successfully eliminate the relevant terms in the power series (given by the images of Weyl algebra generators under the lifted automorphism) to make them into polynomials. Thus one obtains a candidate for the Kontsevich isomorphism. The procedure utilizes the following essential features. First, the Weyl algebra over an algebraically closed field of characteristic zero may be identified with a subalgebra in a certain reduced direct product (reduction modulo infinite prime) of Weyl algebras in positive characteristic -- a fact that allows one to use the theory of Azumaya algebras and is particularly helpful when eliminating the infinite series. Second, the lifting is performed via a direct homomorphism $\Aut W_n\rightarrow \Aut P_n$ which is an isomorphism of the tame subgroups (that such an isomorphism exists is known due to our prior work with Kontsevich) and effectively provides an inverse to it. Finally, the lifted automorphism is the limit (in formal power series topology) of a sequence of lifted tame symplectomorphisms; the fact that any polynomial symplectomorphism has a sequence of tame symplectomorphisms converging to it is our development of the work of D. Anick on approximation and is very recent. (This is joint work with A. Elishev and J.-T. Yu)