Welcome to MathOverflow

Question

4

Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.

Does there necessarily exist a Levi subalgebra of $\mathfrak g$ which is $G$-invariant?

By Levi subalgebra I mean a semisimple complement of the solvable radical, as in the Levi-Malcev theorem. My field is $\mathbb Q$... but if needed I could probably deal with extensions of scalars.

edited Mar 2 2012 at 19:56

asked Mar 2 2012 at 9:02

Mariano Suárez-Alvarez
30.6k●7●68●153

"Levi subalgebra" has more than one meaning nowadays, so it's important to include a precise definition. (For instance, "Levi subalgebra" sometimes means a complement to the nilradical of an arbitrary parabolic subalgebra.) Also, does it matter whether the field is assumed to be algebraically closed? – Jim Humphreys Mar 2 2012 at 13:56

Pasha Zusmanovich · Accepted Answer · 2012-04-04 11:15:24Z UTC

3

Yes. This results and its variations are contained in papers by E.J. Taft:

Invariant Wedderburn factors, Illinois J. Math. 1 (1957), N4, 565-573 http://projecteuclid.org/euclid.ijm/1255380679 .
Invariant Levi factors, Michigan Math. J. 9 (1962), N1, 65-68 DOI:10.1307/mmj/1028998623
Orthogonal conjugacies in associative and Lie algebras, Trans. AMS 113 (1964), No.1, 18-29 DOI:10.2307/1994088

answered Apr 4 2012 at 11:15

Pasha Zusmanovich
1,668●6●12

	Taft's results from 1957+ still seem to be optimal, though I'd forgotten about them. Note that the first paper is most important for the question here, though done in wider generality than Lie algebras. (Also, the other two papers are freely available online, via Project Euclid projecteuclid.org/… or via AMS e-math page.) – Jim Humphreys Apr 4 2012 at 17:32
	Ah! Great! (Thank you for going through my list of questions, by the way :=) ) – Mariano Suárez-Alvarez Apr 4 2012 at 18:08

Equivariant Levi subalgebras.

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

1 Answer

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.