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 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.
T:V\toV
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.[1] [2] 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.[1] 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.[3]
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)
| M | |
| n1 |
(D1) x
| M | |
| n2 |
(D2) x … x
| M | |
| nr |
(Dr)
Di
Mn(D)
An operator T is semi-simple in the sense above if and only if the subalgebra
F[T]\subseteq\operatorname{End}F(V)
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
0\toM'\toM\toM''\to0
M\congM' ⊕ M''
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[4] C is called semi-simple if there is a collection of simple objects
X\alpha\inC
X\alpha
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.[5] Another example from algebraic geometry is the category of pure motives of smooth projective varieties over a field k
\operatorname{Mot}(k)\sim
\sim
Semisimple abelian categories also arise from a combination of a t-structure and a (suitably related) weight structure on a triangulated category.
See main article: Semisimple representation.
See also: Maschke's theorem and Weyl's theorem on complete reducibility.
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
R
\Pi(x)=\begin{pmatrix} 1&x\\ 0&1 \end{pmatrix}
e1
G
\Pi
G
\Pi
\Pi
ak{g}
ak{g}
ak{g}
K
K
K
ak{g}
ak{g}
See also: Fusion category (which are semisimple).
Are abelian non-degenerate tensor categories semisimple?