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 $n\ge 3$, 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.

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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.

Perspectives

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This technique seems to be useful for other types of Ind-schemes, in particular for symplectomorphisms and Weil algebra endomorphisms in spirit of Kontsevich conjecture

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