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In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''simple'' objects, and simple objects are those which do not contain non-trivial sub-objects. The precise definitions of these words depends on the context. For example, if ''G'' is a finite group, then a nontrivial finite-dimensional representation ''V'' over a field is said to be ''simple'' if the only subrepresentations it contains are either or ''V'' (these are also called irreducible representations). Then Maschke's theorem says that any finite-dimensional representation is a direct sum of simple representations (provided the characteristic does not divide the order of the group). So, in this case, every representation of a finite group is ''semi-simple''. Especially in algebra and representation theory, "semi-simplicity" is also called complete reducibility. For example, Weyl's theorem on complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple. A square matrix (in other words a linear operator with ''V'' finite dimensional vector space) is said to be ''simple'' if the only subspaces which are invariant under ''T'' are and ''V''. If the field is algebraically closed (such as the complex numbers), then the only simple matrices are of size 1 by 1. A ''semi-simple matrix'' is one that is similar to a direct sum of simple matrices; if the field is algebraically closed, this is the same as being diagonalizable. These notions of semi-simplicity can be unified using the language of semi-simple modules, and generalized to semi-simple categories. ==Introductory example of vector spaces== If one considers all vector spaces (over a field, such as the real numbers), the simple vector spaces are those which contain no proper subspaces. Therefore, the one-dimensional vector spaces are the simple ones. So it is a basic result of linear algebra that any finite-dimensional vector space is the direct sum of simple vector spaces; in other words, all finite-dimensional vector spaces are semi-simple. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Semi-simplicity」の詳細全文を読む スポンサード リンク
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