# 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. In particular, a*semisimple*representation is*completely reducible*, i.e., is a direct sum of irreducible representations (under a descending chain condition). One speaks of an abelian category as being*semisimple*when every object has the corresponding property.

- A
*semisimple ring*or*semisimple algebra*is one that is semisimple as a module over itself.

- A
*semisimple*matrix is diagonalizable over any algebraically closed field containing its entries. In practice this means that it has a diagonal matrix as its Jordan normal form.

- A
*semisimple Lie algebra*is a Lie algebra which is a direct sum of simple Lie algebras.

- A connected Lie group is called
*semisimple*when its Lie algebra is; and the same for algebraic groups. Every finite dimensional representation of a semisimple Lie algebra, Lie group, or algebraic group in characteristic 0 is semisimple, i.e., completely reducible, but the converse is not true. (See reductive group.) Moreover, in characteristic*p*>0, semisimple Lie groups and Lie algebras have finite dimensional representations which are not semisimple. An element of a semisimple Lie group or Lie algebra is itself*semisimple*if its image in every finite-dimensional representation is semisimple in the sense of matrices.

- A linear algebraic group
*G*is called*semisimple*if the radical of the identity component*G*of^{0}*G*is trivial.*G*is semisimple if and only if*G*has no nontrivial connected abelian normal subgroup.