3 added 172 characters in body

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. 2 added 79 characters in body; added 11 characters in body 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.

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 $k[x,y]$.