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 formula_1 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 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 that 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.
Semi-simple matrices.
A 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 basis for this space can be extended to an eigenbasis.
Semi-simple modules and rings.
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.
Examples of semi-simple rings include fields and, more generally, finite direct products of fields. 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) formula_2, where each formula_3 is a division ring and formula_4 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 subalgebra formula_5 generated by the powers (i.e., iterations) of "T" inside the ring of endomorphisms of "V" is semi-simple.
As indicated above, the theory of semi-simple rings is much more easy than the one of general rings. For example, any short exact sequence
of modules over a semi-simple ring must split, i.e., formula_7. From the point of view of homological algebra, this means that there are no non-trivial extensions. The ring Z of integers is not semi-simple: Z is not the direct sum of "n"Z and Z/"n".
Semi-simple categories.
Many of the above notions of semi-simplicity are recovered by the concept of a "semi-simple" category "C". Briefly, a category is a collection of objects and maps between such objects, the idea being that the maps between the objects preserve some structure inherent in these objects. For example, "R"-modules and "R"-linear maps between them form a category, for any ring "R".
An abelian category "C" is called semi-simple if there is a collection of simple objects formula_8, i.e., ones with no subobject other than the zero object 0 and formula_9 itself, such that "any" object "X" is the direct sum (i.e., coproduct or, equivalently, product) of finitely many simple objects. It follows from Schur's lemma that the endomorphism ring
in a semi-simple category is a product of matrix rings over division rings, i.e., semi-simple.
Moreover, a ring "R" is semi-simple if and only if the category of finitely generated "R"-modules is semisimple.
An example from Hodge theory is the category of "polarizable pure Hodge structures", i.e., pure Hodge structures equipped with a suitable positive definite bilinear form. The presence of this so-called polarization causes the category of polarizable Hodge structures to be semi-simple.
Another example from algebraic geometry is the category of "pure motives" of smooth projective varieties over a field "k" formula_11 modulo an adequate equivalence relation formula_12. As was conjectured by Grothendieck and shown by Jannsen, this category is semi-simple if and only if the equivalence relation is numerical equivalence. This fact is a conceptual cornerstone in the theory of motives.
Semisimple abelian categories also arise from a combination of a t-structure and a (suitably related) weight structure on a triangulated category.
Semi-simplicity in representation theory.
One can ask whether the category of finite-dimensional representations of a group or a Lie algebra is semisimple, that is, whether every finite-dimensional representation decomposes as a direct sum of irreducible representations. The answer, in general, is no. For example, the representation of formula_13 given by
is not a direct sum of irreducibles. (There is precisely one nontrivial invariant subspace, the span of the first basis element, formula_15.) On the other hand, if formula_16 is compact, then every finite-dimensional representation formula_17 of formula_16 admits an inner product with respect to which formula_17 is unitary, showing that formula_17 decomposes as a sum of irreducibles. Similarly, if formula_21 is a complex semisimple Lie algebra, every finite-dimensional representation of formula_21 is a sum of irreducibles. Weyl's original proof of this used the unitarian trick: Every such formula_21 is the complexification of the Lie algebra of a simply connected compact Lie group formula_24. Since formula_24 is simply connected, there is a one-to-one correspondence between the finite-dimensional representations of formula_24 and of formula_21. Thus, the just-mentioned result about representations of compact groups applies. It is also possible to prove semisimplicity of representations of formula_21 directly by algebraic means, as in Section 10.3 of Hall's book.
See also: fusion category (which is semisimple).