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.
Read more about Semisimple Lie Algebra: Examples, Classification, History, Significance, Generalizations
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