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Independence of the B-KK Isomorphism of Infinite Prime

Research paper by Alexei Kanel-Belov, Andrey Elishev

Indexed on: 21 Dec '15Published on: 21 Dec '15Published in: Mathematics - Algebraic Geometry



Abstract

We investigate a certain class of ind-group 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. An open conjecture of Kanel-Belov and Kontsevich states that these automorphism groups are canonically isomorphic in characteristic zero, with the mapping discussed here possibly realizing the isomorphism. The goal 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 some non-principal ultrafilter $\mathcal{U}$ on the set of positive integers. This independence of non-constructible objects issue, which is quite intricate (often resolving negatively in similar settings), is closely related to the famous Dixmier conjecture on Weyl algebra endomorphisms as well as to a number of growth/dimension problems in universal algebraic geometry.