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We show that any normal irreducible affine $n$-dimensional $SL_n$-variety $X$is determined by its automorphism group in the category of normal irreducibleaffine varieties: if $Y$ is an irreducible affine normal algebraic variety suchthat $Aut(X) \cong Aut(Y)$ as ind-groups, then $Y \cong X$ as varieties. If wedrop the condition of normality on $Y$ , then $X$ is not uniquely determinedand we classify all such varieties. In case $n \ge 3$, all the above resultshold true if we replace $Aut(X)$ by $U(X)$, where $U(X)$ is the subgroup of$Aut(X)$ generated by all one-dimensional unipotent subgroups. In dimension $2$we have some very interesting exceptions.

Characterization of $n$-dimensional normal affine $SL_n$ -varieties