Semisimple Lie Algebra

Semisimple Lie Algebra

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras whose only ideals are {0} and itself.

Throughout the article, unless otherwise stated, is a finite-dimensional Lie algebra over a field of characteristic 0. The following conditions are equivalent:

• is semisimple
• the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate,
• has no non-zero abelian ideals,
• has no non-zero solvable ideals,
• The radical of is zero.