11
$\begingroup$

This question is similar to, but not the same as this one. Take the space of automorphisms of $\mathbb{C}^n$ in the holomorphic category, with the compact-open topology. For $n=1$ this is just $\mathbb{C} \times \mathbb{C}^*$, but for larger $n$ is very complicated. It clearly contains $GL(n,\mathbb{C})$, and translations.

To give a taste of how big $Aut(\mathbb{C}^n)$ is, there is a theorem that given any two countable dense subsets $X,Y \subset \mathbb{C}^n$, $n >1$, there is a volume preserving automorphism taking $X$ to $Y$. But I have no idea about what this space is like. Is it some sort of infinite dimensional manifold? Analytic space? Does it contain an infinite dimensional Lie group as a (closed) subgroup?

$\endgroup$
3
  • $\begingroup$ I don't know what counts as an infinite-dimensional Lie group, but you can make lots of closed subgroups, for example the stabilizer of a point or a line or a function $\mathbb C^n\to \mathbb C$. $\endgroup$ Dec 1, 2010 at 3:27
  • $\begingroup$ Say a Frechet Lie group. A Banach Lie group might be too much to ask. $\endgroup$
    – David Roberts
    Dec 1, 2010 at 4:23
  • 5
    $\begingroup$ In the algebraic case it's horrifying: not representable for any $n > 1$. Indeed, suppose rep'td by some $G$. Then Grothendieck's functorial criterion for being locally of finite presentation forces $G$ to be loc. of finite type, then smooth (by Cartier), so $G^0$ is a finite type connected smooth group variety. This forces the points of $G^0$ to act by automorphisms represented by polynomials of bounded degree, which is absurd for $n > 1$ since for two distinct variables $(x,y)$ can use $(x,y) \mapsto (x + t y^n, y)$ to get a connected family of auts joining id to arb. big degree. $\endgroup$
    – BCnrd
    Dec 1, 2010 at 5:06

1 Answer 1

5
$\begingroup$

It is a group generated by flows of holomorphic vector fields. Its Lie algebra is a set of all holomorphic vector fields. As I can for now remember there were a computation of its group of cohomologies (by Feigin and Fuchs: http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=3301&option_lang=eng

http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=intf&paperid=93&option_lang=eng )

Maybe this book can help if you have no translation of these articles from Russian: Fuks D.B. Cohomology of infinite-dimensional Lie algebras, Consultants Bureau, 1986

I think that you have much to learn from this book.

Also there is an article: http://arxiv.org/abs/0708.3398 - it may be rather useful

$\endgroup$
2
  • 2
    $\begingroup$ But you can easily write down two polynomial vector fields that have global flows, and whose sum does not. $\endgroup$ Dec 2, 2010 at 17:33
  • $\begingroup$ @Tom: Maybe. I really don't know if it so. $\endgroup$
    – zroslav
    Dec 2, 2010 at 20:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.