Semisimple Lie algebra

From Wikipedia, the free encyclopedia

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., nonabelian 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 nondegenerate,
$\mathfrak g$ has no nonzero abelian ideals,
$\mathfrak g$ has no nonzero 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.