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.

40 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

References in corpus (1)

• On Planar Algebraic Curves and Holonomic $\mathcal{D}$-modules in Positive Characteristic

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