SEMISIMPLE

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 (or linear transformation of finite-dimensional vector spaces) is one for which every invariant subspace has an invariant complement. This is equivalent to the minimal polynomial having only irreducible factors with multiplicity one. Over a perfect field, this amounts to saying that the matrix has simple roots in the algebraic closure (or any larger algebraically closed field), i.e., it becomes diagonalizable over the algebraic closure. Thus, over an algebraically closed field, “semisimple” and “diagonalizable” are synonymous for matrices.

★ A ''semisimple Lie algebra'' is a Lie algebra which is a direct sum of simple Lie algebras. A Lie algebra $g$ is simple if its dimension is larger than one and if it does not contain any nontrivial ideals. This means that if $I\; subset\; g$ is such that $[x,y]in\; I$ for any $yin\; g$ if $xin\; I$, then $I$ is either zero or the whole Lie algebra.

★ 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 that 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 ''G⁰'' of ''G'' is trivial. ''G'' is semisimple if and only if ''G'' has no nontrivial connected abelian normal subgroup.

★ An abelian category ''A'' is said to be ''semisimple'' if every short exact sequence splits in ''A''. For example, the representation categories of compact groups are semisimple abelian.

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See also

See also

★ simple (algebra)