Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space

We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X{)}^0$ is a torus. Concerning the structure of $Aut(A^n)$ we prove that any homomorphism $Aut(A^n)\to G$ of ind-groups either factors through the Jacobian determinant $jac\text{\hspace{0.17em}⁣}:Aut(A^n)\to k^{\ast }$, or it is a closed immersion. For $SAut(A^n):=\ker (jac)$ we show that every nontrivial homomorphism $SAut(A^n) \to G$ is a closed immersion. Finally, we prove that every non-trivial homomorphism $SAut(A^n)\to SAut(A^n)$ is an automorphism, and is given by conjugation with an element from $Aut(A^n)$.