Talk:Semisimple Lie algebra
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The link to Weyl's theorem currently leads to a disambiguation page. Someone with more knowledge of the subject should correct the link to lead to a specific article.--Bill 19:58, 3 February 2006 (UTC)
- It’s been fixed in the interval – thanks for the note!
- —Nils von Barth (nbarth) (talk) 01:33, 1 December 2009 (UTC)
What is that for a semisimple Lie algebra in general? As I understand, it is defined via (adjoint) representation; that is, first we write ad(x) as a sum of a diagonalizable matrix and a nilpotent matrix. Since ad is injective thanks to the center being trivial, we can write x = y + z so that ad(y) and ad(z) correspond to the semisimple part and the nilpotent part of ad(x). (Incidentally, don't we need alg. closed? Maybe the requirement can be dropped via extension of scalars but this isn't a trivial matter.) A technical issue is that for a linear Lie algebra, this construction may not be canonical. It is only so in the semisimple case (I think). In any case, the sentence in question needs to be expanded. -- Taku (talk) 13:01, 1 December 2009 (UTC)
- I have added a paragraph expounding on the Jordan decomposition. I was careful of course to include the hypothesis of algebraic closure in the statement. (In this case, passing to a field extension is actually trivial since we only need to "go up" rather than "go down": in any event, the statement remains true under these more general conditions.) Sławomir Biały (talk) 13:40, 1 December 2009 (UTC)