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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 that do not contain non-trivial proper 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 {0} or V. Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations. So in the case of finite groups with this condition, every finite-dimensional representation 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 is said to be simple if its only invariant subspaces under T are {0} and V. If the field is algebraically closed , 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.