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 ring is one with a Jacobson radical that is {0}.

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. One speaks of an abelian category as being semisimple when every object has the corresponding property.

A semisimple matrix is diagonalizable over any algebraically closed field containing its entries.

A Lie algebra is called semisimple when it is a direct sum of simple Lie algebras, i.e., non-trivial Lie algebras L whose only ideals are {0} and L itself. An equivalent condition is that the Killing form κ(x,y) = Trace(ad(x) ad(y)) is non-degenerate. 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.

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