Semi-simplicity
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.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.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.For vector spaces over an algebraically closed field F, semi-simplicity of a matrix is equivalent to diagonalizability.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.As it turns out, this is equivalent to requiring that any finitely generated R-module M is semi-simple.For a finite group G Maschke's theorem asserts that the group ring R[G] over some ring R is semi-simple if and only if R is semi-simple and |G| is invertible in R. Since the theory of modules of R[G] is the same as the representation theory of G on R-modules, this fact is an important dichotomy, which causes modular representation theory, i.e., the case when |G| does divide the characteristic of R to be more difficult than the case when |G| does not divide the characteristic, in particular if R is a field of characteristic zero.By the Artin–Wedderburn theorem, a unital Artinian ring R is semisimple if and only if it is (isomorphic to)is the ring of n-by-n matrices with entries in D. An operator T is semi-simple in the sense above if and only if the subalgebragenerated by the powers (i.e., iterations) of T inside the ring of endomorphisms of V is semi-simple.For example, any short exact sequence of modules over a semi-simple ring must split, i.e.,From the point of view of homological algebra, this means that there are no non-trivial extensions.Moreover, a ring R is semi-simple if and only if the category of finitely generated R-modules is semisimple.The presence of this so-called polarization causes the category of polarizable Hodge structures to be semi-simple.[5] Another example from algebraic geometry is the category of pure motives of smooth projective varieties over a field k[8] (There is precisely one nontrivial invariant subspace, the span of the first basis element,is a complex semisimple Lie algebra, every finite-dimensional representation of[11] Thus, the just-mentioned result about representations of compact groups applies.