Lifting of polynomial symplectomorphisms and deformation quantization

What is it about?

The lifting problem has its origins in the context of deformation quantization of the affine space and is closely related to several major open problems in algebraic geometry and ring theory. The lifting problem is the milestone in the Kontsevich conjecture. The conjecture states: The automorphism groups of the n-th Weyl algebra and the polynomial algebra in 2n variables with Poisson structure over the rational numbers are isomorphic. The focus of the paper under review is the problem of lifting of symplectomorphisms: Can an arbitrary symplectomorphism in dimension 2n be lifted to an automorphism of the n-th Weyl algebra in characteristic zero? It establishes the approximation property for polynomial symplectomorphisms and comments on the lifting problem of polynomial symplectomorphisms and Weyl algebra automorphisms. The paper proves the possibility of lifting a symplectomorphism to an automorphism of the power series completion of the Weyl algebra of the corresponding rank. It studies the problem of lifting of polynomial symplectomorphisms in characteristic zero to automorphisms of the Weyl algebra by means of approximation by tame automorphisms. The use of tame approximation is advantageous due to the fact that tame symplectomorphisms correspond to Weyl algebra automorphisms. Reviewed by Vida Milani

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