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On the augmentation topology of automorphism groups of affine spaces and algebras

Alexei Kanel-Belov

Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel

E-mail Address: beloval@cs.biu.ac.il

E-mail Address: kanelster@gmail.com

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Jie-Tai Yu

College of Mathematics and Statistics, Shenzhen University, Shenzhen 518061, P. R. China

E-mail Address: jietaiyu@szu.edu.cn

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Andrey Elishev

Laboratory of Advanced Combinatorics and Network Applications, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia

E-mail Address: elishev@phystech.edu

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https://doi.org/10.1142/S0218196718400040Cited by:8 (Source: Crossref)

This article is part of the issue:

Special Issue on Groups, Algebras and Identities in Honor of Professor Boris Plotkin's 90th Birthday; Guest Editor: E. Plotkin

Abstract

We study topological properties of Ind-groups $Aut (K [x_{1}, \dots, x_{n}])$ $Aut (K [x_{1}, \dots, x_{n}])$ and $Aut (K ⟨ x_{1}, \dots, x_{n} ⟩)$ $Aut (K 〈 x_{1}, \dots, x_{n} 〉)$ of automorphisms of polynomial and free associative algebras via Ind-schemes, toric varieties, approximations, and singularities. We obtain a number of properties of $Aut (Aut (A))$ $Aut (Aut (A))$ , where $A$ $A$ is the polynomial or free associative algebra over the base field $K$ $K$ . We prove that all Ind-scheme automorphisms of $Aut (K [x_{1}, \dots, x_{n}])$ $Aut (K [x_{1}, \dots, x_{n}])$ are inner for $n \geq 3$ $n \geq 3$ , and all Ind-scheme automorphisms of $Aut (K ⟨ x_{1}, \dots, x_{n} ⟩)$ $Aut (K 〈 x_{1}, \dots, x_{n} 〉)$ are semi-inner. As an application, we prove that $Aut (K [x_{1}, \dots, x_{n}])$ $Aut (K [x_{1}, \dots, x_{n}])$ cannot be embedded into $Aut (K ⟨ x_{1}, \dots, x_{n} ⟩)$ $Aut (K 〈 x_{1}, \dots, x_{n} 〉)$ by the natural abelianization. In other words, the Automorphism Group Lifting Problem has a negative solution. We explore close connection between the above results and the Jacobian conjecture, as well as the Kanel-Belov–Kontsevich conjecture, and formulate the Jacobian conjecture for fields of any characteristic. We make use of results of Bodnarchuk and Rips, and we also consider automorphisms of tame groups preserving the origin and obtain a modification of said results in the tame setting.

Communicated by E. Plotkin

Dedicated to Prof. B.I. Plotkin who initiated the study of this subject.

Keywords:

AMSC: 13F20, 16S10, 14R10, 16W20, 16Z05

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Vol. 28, No. 08

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History

Received 25 November 2017

Accepted 28 June 2018

Published: 23 August 2018

Keywords

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Augmentation topology of infinitely dimensional Ind-schemes in connection with their authmorphisms.

What is it about?

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 $nge 3$, and all $Ind$-scheme automorphisms of $Aut(Klangle x_1,dots, x_n angle)$ are semi-inner. As an application, we prove that $Aut(K[x_1,dots,x_n])$ cannot be embedded into $Aut(Klangle x_1,dots,x_n angle)$ 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.

Why is it important?

Naive approach to Ind-schemes due to singularity effects fails. We overcame this effect using parametric curves and preserving infinity trick. This is important tool for Ind-schemes theory.

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Alexey Kanel-Belov

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On the augmentation topology of automorphism groups of affine spaces and algebras

Abstract

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