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

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 $Aut(K⟨x_1,\dots ,x_n⟩)$ are semi-inner. As an application, we prove that $Aut(K[x_1,\dots ,x_n])$ cannot be embedded into $Aut(K⟨x_1,\dots ,x_n⟩)$ 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.

arXiv comment40 pages, dedicated to 90-th aniversary of prof. B.I.Plotkin. arXiv admin note: substantial text overlap with arXiv:1207.2045, Acctped to the special Issue of International Journal of Aljebra and computation, dedicated to prof. B.I.Plotkin, 2018

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• Automorphisms of the Weyl algebra
• Automorphisms of the endomorphism semigroup of a free commutative algebra
• Automorphisms of the semigroup of endomorphisms of free associative algebras
• On the lifting of the Nagata automorphism
• Stable tameness of automorphisms of F<x,y,z> fixing z
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Cited by in corpus (5)

• Infinite transitivity, finite generation, and Demazure roots
• Polynomial automorphisms, quantization and Jacobian conjecture related problems
• Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry
• Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems
• Noncommutative Bialynicki-Birula Theorem