Welcome to MathOverflow

Question

9

2

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 $k$?

edited Jan 15 at 17:39

asked Jan 15 at 16:12

Mikhail Borovoi
2,672●6●14

1

What about inversion (for $n>1$)? – ACL Jan 15 at 16:27

@ACL: Thank you, Antoine. Are there any other automorphisms? – Mikhail Borovoi Jan 15 at 16:33

@Mikhail: Your edit needs some more editing. Aside from this, what motivates the original question? – Jim Humphreys Jan 15 at 17:04

@Jim: I have removed the edit. The original question was motivated by my previous question mathoverflow.net/questions/118356/… and a comment of Tom Goodwillie. I am trying to construct a finite subgroup $H\subset G=SL_{n,\mathbb{C}}$ and an automorphism $\sigma$ of $\mathbb{C}$ such that the $\mathbb{C}$-varieties $G/H$ and $\sigma(G/H)=G/\sigma H$ are not isomorphic. – Mikhail Borovoi Jan 15 at 17:50

Note that the group generated by automorphisms, left translations and inversion is finite-dimensional (actually $2(n^2-1)$); while the example by Mariano gives a faithful action of an infinite dimensional abelian group. – Yves Cornulier Jan 15 at 18:10

Mariano Suárez-Alvarez · Accepted Answer · 2013-01-15 17:05:03Z UTC

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.

Jason Starr · Answer 2 · 2013-01-15 19:45:09Z UTC

9

The automorphism group is massive!

Flexible varieties and automorphism groups, I. Arzhantsev, H. Flenner, S. Kaliman, F. Kutzschebauch, M. Zaidenberg, http://arxiv.org/abs/1011.5375.

edited Jan 15 at 19:45

Jason Starr

community wiki

2 revisions

Automorphisms of $SL_n$ as a variety

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.