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

flag	asked Apr 3 2011 at 21:53 mark 53●2

Faisal · Accepted Answer · 2011-04-03 22:13:08Z UTC

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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 $\text{Rad}(L \times V) = V$ .

answered Apr 3 2011 at 22:13

Faisal
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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 2011 at 13:08

lie algebra semisimple?

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