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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., nontrivial Lie algebras g whose only ideals are {0} and g itself.
Let g be a Lie algebra. The following conditions are equivalent:
- g is semisimple,
- the Killing form, κ(x,y) = tr(ad(x)ad(y), is nondegenerate,
- g has no nontrivial abelian ideals,
- g has no nontrivial solvable ideals,
- the radical of g is 0.
When g is defined over a field of characteristic zero, g is semisimple if and only if every representation is completely reducible, that is for every invariant subspace of the representation there is an invariant complement (Weyl's theorem).
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