Semisimple Lie Algebra
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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras g whose only ideals are {0} and g itself. A consequence of semisimplicity is a theorem due to Weyl: every finite-dimensional representation is completely reducible; that is for every invariant subspace of the representation there is an invariant complement. While in other contexts, complete reducibility is equivalent to being semisimple, for Lie algebras the two notions are different: Lie…mehr

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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles! In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras g whose only ideals are {0} and g itself. A consequence of semisimplicity is a theorem due to Weyl: every finite-dimensional representation is completely reducible; that is for every invariant subspace of the representation there is an invariant complement. While in other contexts, complete reducibility is equivalent to being semisimple, for Lie algebras the two notions are different: Lie algebras whose finite-dimensional representations are all completely reducible are called reductive Lie algebras.