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 sub-objects. The precise definitions of these words depends on the context.
Contents
- Introductory example of vector spaces
- Semi simple matrices
- Semi simple modules and rings
- Semi simple categories
- Semi simplicity in representation theory
- References
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). Then Maschke's theorem says that any finite-dimensional representation is a direct sum of simple representations (provided the characteristic does not divide the order of the group). So, in this case, every representation of a finite group 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
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)
An operator T is semi-simple in the sense above if and only if the subalgebra
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.,
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
in a semi-simple category is a product of division algebras, 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
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 (say, finite-dimensional) representations of a group G is semisimple or not (in such a category, irreducible representations are precisely simple objects). For example, the category is semisimple if G is a semisimple compact Lie group (Weyl's theorem on complete reducibility).
See also: fusion category (which is semisimple).