Semi-simplicity

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 (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 ${\displaystyle T:V\to V}$ with V a finite-dimensional vector space) is said to be simple if its only invariant linear 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 categories.

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. 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.

A square matrix or, equivalently, a linear operator T on a finite-dimensional vector space V is called semi-simple if every T-invariant subspace has a complementary T-invariant subspace. This is equivalent to the minimal polynomial of T being square-free.

For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability. This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane, which itself has an eigenvector, and thus by induction is diagonalizable. Conversely, diagonalizable operators are easily seen to be semi-simple, as invariant subspaces are direct sums of eigenspaces, and any eigenbasis for this subspace can be extended to an eigenbasis of the full space.

For a fixed ring R, a nontrivial R-module M is simple, if it has no submodules other than 0 and M. An R-module M is semi-simple if every R-submodule of M is an R-module direct summand of M (the trivial module 0 is semi-simple, but not simple). For an R-module M, M is semi-simple if and only if it is the direct sum of simple modules (the trivial module is the empty direct sum). Finally, R is called a semi-simple ring if it is semi-simple as an R-module. As it turns out, this is equivalent to requiring that any finitely generated R-module M is semi-simple.