On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groups

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 $Ind$-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 $Aut(Aut(A))$, where $A$ is the polynomial or free associative algebra over the base field $K$. We prove that all $Ind$-scheme automorphisms of $Aut(K[x_1,\dots,x_n])$ are inner for $n\ge 3$, and all $Ind$-scheme automorphisms of $Aut(K\langle x_1,\dots, x_n\rangle)$ are semi-inner. As an application, we prove that $Aut(K[x_1,\dots,x_n])$ cannot be embedded into $Aut(K\langle x_1,\dots,x_n\rangle)$ 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.