Automorphism groups of affine varieties and a characterization of affine n-space

We show that the automorphism group of affine  -space determines up to isomorphism: If  is a connected affine variety such that as ind-groups, then as varieties. We also show that every torus appears as for a suitable irreducible affine variety  , but that cannot be isomorphic to a semisimple group. In fact, if is finite-dimensional and if , then the connected component is a torus. Concerning the structure of we prove that any homomorphism of ind-groups either factors through where is the Jacobian determinant, or it is a closed immersion. For we show that every nontrivial homomorphism is a closed immersion. Finally, we prove that every nontrivial homomorphism is an automorphism, and that  is given by conjugation with an element from .