SEMISIMPLE LIE ALGEBRA

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.

Contents

See also

See also

★ semisimple

★ simple Lie algebra

★ reductive group