Semisimple

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In mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way (by direct sum).

• A semisimple module is one in which each submodule is a direct summand
• A semisimple algebra (or ring) is one that is semisimple as a module over itself
• A semisimple operator (or matrix) is one for which every invariant subspace has an invariant complement. This is equivalent to the minimal polynomial being square-free. Over an algebraically closed field it is equivalent to diagonalizable.
• A semisimple Lie algebra is a Lie algebra which is a direct sum of simple Lie algebras.
• A semisimple algebraic group is a linear algebraic group whose radical of the identity component is trivial

[edit] See also

• simple (algebra)

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