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

Kanel-Belov, A.; Jie-Tai, Y.; Elishev, A.

Авторы: Kanel-Belov A., Jie-Tai Yu, Elishev A.
Сборник: ArXiv.org.e-Print
Серия: [math.PR]
Том: 1708
Год издания: 2017
Издательство: Cornell University Press
Местоположение издательства: United States
Аннотация: 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. Comments: 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
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On the tame authomorphism approximation, augmentation Topology of Automorphism Groups and $Ind$-schemes, and authomorphisms of tame automorphism groupsстатья

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