Semisimple Lie Algebra

A Lie algebra over a field of characteristic zero is called semisimple if its Killing form is nondegenerate. The following properties can be proved equivalent for a finite-dimensional algebra L over a field of characteristic 0:

1. L is semisimple.

2. L has no nonzero Abelian ideal.

3. L has zero ideal radical (the radical is the biggest solvable ideal).

4. Every representation of L is fully reducible, i.e., is a sum of irreducible representations.

5. L is a (finite) direct product of simple Lie algebras (a Lie algebra is called simple if it is not Abelian and has no nonzero ideal !=L).

See also

Semisimple Lie Group, Simple Lie Algebra

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Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations. New York: Springer-Verlag, 1984.

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Semisimple Lie Algebra

Cite this as:

Weisstein, Eric W. "Semisimple Lie Algebra." From MathWorld--A Wolfram Web Resource.

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