[Paper Review] Independence of the B-KK Isomorphism of Infinite Prime

Alexei Kanel-Belov, Andrey Elishev+1 more|arXiv (Cornell University)|Dec 21, 2015

Advanced Differential Equations and Dynamical Systems38 references3 citations

TL;DR

This paper establishes the independence of the B-KK isomorphism — a canonical isomorphism between the automorphism groups of the Weyl algebra and the Poisson algebra — from the choice of infinite prime in characteristic zero. By introducing augmented and skew-augmented algebras and employing tame approximation with singularity analysis in Ind-schemes, the authors prove that the isomorphism is well-defined and independent of the ultrafilter choice, confirming a key step toward the full Kontsevich conjecture on the isomorphism of automorphism groups.

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.

Motivation & Objective

• To establish the independence of the B-KK isomorphism from the choice of infinite prime, i.e., the class $[p]$ of prime sequences modulo a non-principal ultrafilter.
• To resolve a central technical obstacle in the proof of the Kontsevich conjecture by showing that the isomorphism between automorphism groups of the Weyl and Poisson algebras is not dependent on the model-theoretic construction of infinite primes.
• To extend the lifting map from automorphisms to the completion of skew Weyl algebras using a deformation parameter $h$, enabling a well-behaved inverse to the homomorphism $\phi_{[p]}$.
• To demonstrate that the specialization of augmentation parameters in the skew-augmented setting preserves the isomorphism structure, thereby enabling a robust approximation argument.

Proposed method

• Introduce augmented and skew-augmented versions of the Weyl and Poisson algebras to control the behavior of automorphisms under limits and specialization.
• Construct an augmented Ind-scheme morphism between normalized automorphism Ind-schemes, using the deformation parameter $h$ to stabilize the lifting map.
• Analyze singularities of curves in skew-augmented automorphism Ind-schemes and their images under Ind-scheme morphisms to ensure proper approximation.
• Apply a specialization argument that switches from $h$-preserving automorphisms to rational functions over $\mathbb{C}(h)$, leveraging results from Nagata and others.
• Use homotopy and irreducibility arguments to show that the isomorphism lifts correctly to $h=1$, ensuring the map is well-defined at the classical level.
• Leverage the fact that tame automorphisms in the skew-augmented setting form a dense subset, allowing convergence and independence from the ultrafilter choice.

Experimental results

Research questions

• RQ1 Is the B-KK isomorphism between $\operatorname{Aut}W_{n,\mathbb{C}}$ and $\operatorname{Aut}P_{n,\mathbb{C}}$ independent of the choice of infinite prime $[p]$?
• RQ2 Can the lifting map from symplectomorphisms to Weyl algebra automorphisms be extended to a well-defined isomorphism over $\mathbb{C}[h,h^{-1}]$ in the skew-augmented setting?
• RQ3 Does the specialization of the augmentation parameter $h$ to 1 preserve the isomorphism structure, and is this process independent of the ultrafilter?
• RQ4 Can tame approximation in the augmented setting be used to control the behavior of automorphisms under limits, ensuring independence from infinite prime construction?
• RQ5 Is the endomorphism counterpart of the B-KK isomorphism algebraic, and does it satisfy a similar independence property?

Key findings

• The B-KK isomorphism is independent of the choice of infinite prime $[p]$, as the construction of $\phi_{[p]}$ does not depend on the specific ultrafilter used.
• The introduction of the deformation parameter $h$ allows the lifting map to be extended to an isomorphism over $\mathbb{C}[h,h^{-1}]$, ensuring compatibility with the inverse construction.
• Singularities in skew-augmented automorphism Ind-schemes are controlled via the singularity trick, enabling a robust tame approximation argument.
• The specialization of $h$ to 1 is valid and well-defined, as the isomorphism is continuous and preserves irreducible components under limit processes.
• The augmented and skew-augmented frameworks allow the construction of a well-behaved inverse to $\phi_{[p]}$, confirming the isomorphism is canonical and independent of model-theoretic choices.
• The results extend to arbitrary fields of characteristic zero, as the construction is insensitive to the base field as long as it is characteristic zero.