Semisimple Lie algebra

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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras $\mathfrak g$ whose only ideals are {0} and $\mathfrak g$ itself. It is called reductive if it is the sum of a semisimple and an abelian Lie algebra.

Let $\mathfrak g$ be a finite-dimensional Lie algebra. The following conditions are equivalent:

$\mathfrak g$ is direct sum of simple Lie algebras,
the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate,
$\mathfrak g$ has no non-zero abelian ideals,
$\mathfrak g$ has no non-zero solvable ideals,
the radical of $\mathfrak g$ is 0.

Additionally, when $\mathfrak g$ is defined over a field of characteristic 0 we have:

$\mathfrak g$ 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).

If $\mathfrak g$ is semisimple, then every element can be expressed as the bracket of two other elements, i.e. $\mathfrak g = [\mathfrak g, \mathfrak g]$ . The converse of this statement does not always hold.

[edit] References

Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9