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If L is a semisimple lie algebra then L=[L,L]. Is the opposite true?

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up vote 21 down vote accepted

No. A Lie algebra satisfying that property is called perfect. For an example of a perfect Lie algebra that isn't semisimple, take a semisimple $L$ and an irreducible representation $V$ of $L$, and define a bracket on $L \times V$ by $$ [(X,v),(Y,u)] := ([X,Y],Xu-Yv). $$ This turns $L \times V$ into a perfect Lie algebra with $\text{Rad}(L \times V) = V$.

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Maybe it's also helpful to mention the substantial though often isolated literature on perfect Lie algebras and related structure theory? There are some interesting connections with other questions, as in the paper by Benkart and Zelmanov in Invent. Math. 126 (1996), 1-45. – Jim Humphreys Apr 5 '11 at 13:08
@JimHumphreys, you mention the substantial literature on this subject—where else would one look? – L Spice Jun 11 '15 at 18:49
@LSpice: Besides the paper I mentioned, there are many others listed on MathSciNet under "perfect Lie algebra" though I'm not sure what would interest you. I guess my point was that the purely algebraic theory of Lie algebras (often in characteristic 0) has been studied by many people over the past century; thus, much is known. A random example is the paper by Baranov and Zalesskii: Plain representations of Lie algebras, J. London Math. Soc. (2) 63 (2001), no. 3, 571–591. You can also find some related posts here by searching MO for "perfect Lie algebra". – Jim Humphreys Jun 11 '15 at 21:13
@JimHumphreys, thanks! I didn't mean to ask you to do the search for me; I just thought you might have some pointers to good entrées to this literature. (My approach is that of someone who knows Lie algebras only as things coming from (algebraic) Lie groups, so basically I am interested in anything about them that sheds light on structure theory in that setting (but, unfortunately for the history you mention, particularly in positive characteristic).) – L Spice Jun 11 '15 at 21:36

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