What are the automorphisms of $SL\_n$ as an algebraic variety?


In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL\_n$ regarded as an *algebraic variety* over $k$. Assume that $\tau$ takes the unit element $e$ of $G$ to itself. Is it true that $\tau$ is an automorphism of $SL\_n$ as an *algebraic group* over $k$?



Automorphisms of $SL_n$ as a variety
The coordinate ring when $n=2$ is $A=k[a,b,c,d]/(ad-bc-1)$. 


If $f\in k[b,c]$, there is an automorphism $\phi:A\to A$ such that $\phi(a)=a+bf$, $\phi(c)=c+df$, $\phi(b)=b$ and $\phi(d)=d$. 


One could conjecture that the automorphism group in this case is generated by $SL\_2$, inversion and this sort of triangular automorphisms, much as in the Makar-Limanov–Jung–van der Kulk theorem for $k[x,y]$ (This is a *very* optimistic conjecture, though: this is a $3$-dimensional affine variety quite close to affine space and there are non-tame automorphisms of the latter...)


In general, I doubt we know the automorphism group.


