Semisimple Lie algebra
Direct sum of simple Lie algebras / From Wikipedia, the free encyclopedia
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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals.)
Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such a Lie algebra , if nonzero, 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 (maximal solvable ideal) of is zero.