Automorphisms of $SL_n$ as a variety - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T22:08:59Z http://mathoverflow.net/feeds/question/118988 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118988/automorphisms-of-sl-n-as-a-variety Automorphisms of $SL_n$ as a variety Mikhail Borovoi 2013-01-15T16:12:22Z 2013-01-15T19:45:09Z <p>What are the automorphisms of $SL_n$ as an algebraic variety?</p> <p>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 <em>algebraic variety</em> 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 <em>algebraic group</em> over $k$?</p> http://mathoverflow.net/questions/118988/automorphisms-of-sl-n-as-a-variety/118992#118992 Answer by Mariano Suárez-Alvarez for Automorphisms of $SL_n$ as a variety Mariano Suárez-Alvarez 2013-01-15T16:37:43Z 2013-01-15T17:05:03Z <p>The coordinate ring when $n=2$ is $A=k[a,b,c,d]/(ad-bc-1)$. </p> <p>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 $\phi(d)=d$. </p> <p>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 <em>very</em> optimistic conjecture, though: this is a $3$-dimensional affine variety quite close to affine space and there are non-tame automorphisms of the latter...)</p> <p>In general, I doubt we know the automorphism group.</p> http://mathoverflow.net/questions/118988/automorphisms-of-sl-n-as-a-variety/119011#119011 Answer by Jason Starr for Automorphisms of $SL_n$ as a variety Jason Starr 2013-01-15T18:41:19Z 2013-01-15T19:45:09Z <p>The automorphism group is <B>massive</B>! </p> <p><I>Flexible varieties and automorphism groups</I>, I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, M. Zaidenberg, <a href="http://arxiv.org/abs/1011.5375" rel="nofollow">http://arxiv.org/abs/1011.5375</a>.</p>