Independence of the B-KK Isomorphism of Infinite Prime

Abstract

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 $2n$ variables. A conjecture of Kanel-Belov and Kontsevich, whose proof we have recently obtained, states that these automorphism groups are canonically isomorphic in characteristic zero, with the mapping discussed here being the candidate for the isomorphism. The main 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 $\mathcal{U}$ on the index set of positive integers. To that end, we introduce the augmented and skew 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 skew augmented automorphism Ind-schemes and their images under Ind-scheme morphisms. Apart from that, we study the augmented version of the independence conjecture.