This is work in progress. If N is the nilradical of a solvable Lie algebra then its centraliser, C_L(N), is contained in N. The problem is to find a generalised version of the nilradical, N*, such that C_L(N*), is contained in N* for any Lie algebra. Following the group theory approach seems to work for characteristic zero, but the characteristic p case is more interesting. I've tried a number of modifications but all flounder at some point. Of course, the lack of Levi's Theorem and decomposition of semisimples into simples may be insuperable hurdles; Block's Theorem doesn't seem to give enough. Also the nilradical isn't necessarily characteristic in characteristic p.
I tried replacing 'quasi-simple' with 'quasi-minimal ideal' (two different definitions: an ideal A of L is quasi-minimal in L if A/Z(A) is a minimal ideal of L/Z(A) and A^2 = A; or the same but with the modification 'a minimal ideal of L/Z(A) containing no abelian subideals of L/Z(A)'. Much goes through again, but not the whole theory.
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I have uploaded a revised version of this ( which is still not complete, but has a slightly different approach and eliminates some errors).
I've now uploaded a further version of this paper which incorporates both of the previous approaches. It would be nice to have examples of Lie algebras for which the various radicals are different, but these are not easy to construct. Any ideas would be welcome.
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