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Concept# Semisimple Lie algebra

Summary

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 \mathfrak g, if nonzero, the following conditions are equivalent:
*\mathfrak g is semisimple;
*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 (maximal solvable ideal) of \mathfrak g is zero.

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