We investigate a certain class of Ind-scheme morphisms corresponding to homomorphisms between the automorphism group of the n-th complex Weyl algebra and the group of Poisson structure-preserving automorphisms of the commutative complex polynomial algebra in 2nvariables. An open conjecture of Kanel-Belov and Kontsevich states that these automorphism groups are canonically isomorphic in characteristic zero, with the mapping discussed here being the candidate for the isomorphism. The objective of the present paper is to establish the independence of the said mapping of the choice of infinite prime -that is, the class [p] of prime number sequences modulo fixed non-principal ultrafilter U on the index set of positive integers. To that end, we introduce the augmented versions of algebras in question and study the augmented Ind-morphism between the normalized automorphism Ind-schemes in the context of tame automorphism approximation. In order to correctly implement approximation in our proof, we study singularities of curves in augmented automorphism Ind-schemes and their images under Ind-scheme morphisms.Conjecture 1.1.Aut(A n,C ) ≃ Aut(P n,C ). Here A n,C is the n-th Weyl algebra over the field of complex numbers A n,C = C x 1 , . . . , x n , y 1 , . . . , y n /(x i x j − x j x i , y i y j − y j y i , y i x j − x j y i − δ ij ), and P n,C ≃ C[z 1 , . . . , z 2n ] is the commutative polynomial ring viewed as a C-algebra and equipped with the standard Poisson bracket: {z i , z j } = ω ij ≡ δ i,n+j − δ i+n,j 2010 Mathematics Subject Classification: 14R10
In this paper we study a correspondence between cyclic modules over the first Weyl algebra and planar algebraic curves in positive characteristic. In particular, we show that any such curve has a preimage under a morphism of certain ind-schemes. This property might pave the way to an indirect proof of existence of a canonical isomorphism between the group of algebra automorphisms of the first Weyl algebra over the field complex numbers and the group of polynomial symplectomorphisms of [Formula: see text].
The objective of this paper is the proof of a conjecture of Kontsevich [1] on the isomorphism between groups of polynomial symplectomorphisms and automorphisms of the corresponding Weyl algebra in characteristic zero. The proof is based on the study of topological properties of automorphism Ind-varieties of the so-called augmented and skew augmented versions of Poisson and Weyl algebras. Approximation by tame automorphisms as well as a certain singularity analysis procedure is utilized in the construction of the lifting of augmented polynomial symplectomorphisms, after which specialization of the augmentation parameter is performed in order to obtain the main result.
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