Semisimple Lie algebra
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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. It is called reductive if it is the sum of a semisimple and an abelian Lie algebra.
Let be a finite-dimensional Lie algebra. The following conditions are equivalent:
- is direct sum of simple Lie algebras,
- 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 0.
Additionally, when is defined over a field of characteristic 0 we have:
- 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 is semisimple, then every element can be expressed as the bracket of two other elements, i.e. . The converse of this statement does not always hold.
[edit] References
- Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
- Varadarajan, V. S. Lie Groups, Lie Algebras, and Their Representations, 1st edition, Springer, 2004. ISBN 0-387-90969-9