On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups
A. Kanel-Belov, J.-T. Yu, A. Elishev

TL;DR
This paper investigates automorphisms of automorphism groups of polynomial and free associative algebras, proving innerness or semi-innerness of certain automorphisms and addressing the automorphism group lifting problem, with implications for the Jacobian conjecture.
Contribution
It establishes that all ind-scheme automorphisms of polynomial automorphism groups are inner or semi-inner and demonstrates the non-embeddability of polynomial automorphism groups into free associative automorphism groups.
Findings
All ind-scheme automorphisms of Aut(K[x_1,...,x_n]) are inner for n≥3.
All ind-scheme automorphisms of Aut(K⟨x_1,...,x_n⟩) are semi-inner.
The automorphism group lifting problem has a negative solution.
Abstract
We study authomorphisms of -groups of polynomial automorphisms (wich are singular) via tame approximations. Such objects were pioneeered in research by B.I.Plotkin We obtain a number of properties of , where is the polynomial or free associative algebra over the base field . We prove that all -scheme automorphisms of are inner for , and all -scheme automorphisms of are semi-inner. As an application, we prove that cannot be embedded into by the natural abelianization. In other words, the {\it Automorphism Group Lifting Problem} has a negative solution. We explore close connection between the above results and the Jacobian conjecture type questions, formulate the Jacobian conjecture for fields of any characteristic.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Microtubule and mitosis dynamics · Algebraic Geometry and Number Theory
