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Concept
Semi-simplicity
Résumé
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation 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 (these are also called irreducible representations). Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic of the base field does not divide the order of the group). 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 (in other words a linear operator with V finite dimensional vector space) is said to be simple if its only invariant subspaces under T are {0} 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 .
If one considers all vector spaces (over a field, such as the real numbers), the simple vector spaces are those that contain no proper nontrivial subspaces. Therefore, the one-dimensional vector spaces are the simple ones.
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