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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 $k[x,y]$.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.
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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 $SL_2$ SL_2$, inversion and this sort of triangular automorphisms, much as in the Makar-Limanov–Jung–van der Kulk theorem for $k[x,y]$.

In general, I doubt we know the automorphism group.
show/hide this revision's text 1

The coordinate ring 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 is generated by $SL_2$ and this sort of triangular automorphisms, much as in the Makar-Limanov–Jung–van der Kulk theorem for $k[x,y]$.