In mathematics, semi-simplicity is a widespread concept in disciplines such as
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrices.
...
,
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
,
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
,
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
, and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. 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
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
, then a nontrivial finite-dimensional
representation ''V'' over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
is said to be ''simple'' if the only subrepresentations it contains are either or ''V'' (these are also called
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W,W ...
s). Now
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
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 In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let \mathfrak be a semisimple Lie algebra over a field o ...
says a finite-dimensional representation of a
semisimple
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 ''sim ...
compact
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
is semisimple.
A
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
(in other words a linear operator
with ''V'' finite dimensional vector space) is said to be ''simple'' if its only invariant subspaces under ''T'' are and ''V''. If the field is
algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
(such as the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s), 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
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
.
These notions of semi-simplicity can be unified using the language of semi-simple
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, and generalized to semi-simple
categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
*Categories (Aristotle), ''Categories'' (Aristotle)
*Category (Kant)
...
.
Introductory example of vector spaces
If one considers all
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s (over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
, such as the real numbers), the simple vector spaces are those that contain no proper nontrivial subspaces. Therefore, the one-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
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
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
of simple vector spaces; in other words, all finite-dimensional vector spaces are semi-simple.
Semi-simple matrices
A
square matrix
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
or, equivalently, a
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
''T'' on a finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
''V'' is called ''semi-simple'' if every ''T''-
invariant subspace In mathematics, an invariant subspace of a linear mapping ''T'' : ''V'' → ''V '' i.e. from some vector space ''V'' to itself, is a subspace ''W'' of ''V'' that is preserved by ''T''; that is, ''T''(''W'') ⊆ ''W''.
General descrip ...
has a
complementary
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
''T''-invariant subspace.
[Lam (2001), p. 39/ref> This is equivalent to the minimal polynomial of ''T'' being square-free.
For vector spaces over an ]algebraically closed
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
field ''F'', semi-simplicity of a matrix is equivalent to diagonalizability
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
. This is because such an operator always has an eigenvector; if it is, in addition, semi-simple, then it has a complementary invariant hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
, 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.
Semi-simple modules and rings
For a fixed ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''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
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 ''sim ...
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
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensio ...
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
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make gener ...
asserts that the group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the give ...
''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
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of ''G'' on ''R''-modules, this fact is an important dichotomy, which causes modular representation theory
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as ...
, 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 In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are na ...
''R'' is semisimple if and only if it is (isomorphic to) , where each is a division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element us ...
and 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 generated by the powers (i.e., iterations) of ''T'' inside the ring of endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...
s 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
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
:
of modules over a semi-simple ring must split, i.e., . From the point of view of homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, this means that there are no non-trivial extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
s. 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
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
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
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
''C'' is called semi-simple if there is a collection of simple objects , i.e., ones with no subobject In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subsets from set theory, subgroups from group theory,M ...
other than the zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
0 and itself, such that ''any'' object ''X'' is the direct sum
The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
(i.e., coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
or, equivalently, product) of finitely many simple objects. It follows from Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ' ...
that the endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a p ...
:
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
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohom ...
is the category of ''polarizable pure Hodge structure
In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth and compact Kähler manifold. Hodge structure ...
s'', i.e., pure Hodge structures equipped with a suitable positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
. 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
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
projective varieties
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
over a field ''k'' modulo an adequate equivalence relation In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined i ...
. As was conjectured by Grothendieck and shown by Jannsen, this category is semi-simple if and only if the equivalence relation is numerical equivalence
Numerical may refer to:
* Number
* Numerical digit
* Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distin ...
. 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
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
on a triangulated category In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy categ ...
.
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 given by
:
is not a direct sum of irreducibles. (There is precisely one nontrivial invariant subspace, the span of the first basis element, .) On the other hand, if is compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
, then every finite-dimensional representation of admits an inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
with respect to which is unitary, showing that decomposes as a sum of irreducibles. Similarly, if is a complex semisimple Lie algebra, every finite-dimensional representation of is a sum of irreducibles. Weyl's original proof of this used the unitarian trick
In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some g ...
: Every such is the complexification of the Lie algebra of a simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
compact Lie group . Since is simply connected, there is a one-to-one correspondence between the finite-dimensional representations of and of .[ Theorem 5.6] Thus, the just-mentioned result about representations of compact groups applies. It is also possible to prove semisimplicity of representations of directly by algebraic means, as in Section 10.3 of Hall's book.
See also: Fusion category
In mathematics, a fusion category is a category that is rigid, semisimple, k-linear, monoidal and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field k is algebraically closed ...
(which are semisimple).
See also
*A semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra i ...
is a Lie algebra that is a direct sum of simple Lie algebras.
*A semisimple algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
is a linear algebraic group whose radical of the identity component is trivial.
*Semisimple algebra
In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensio ...
*Semisimple representation
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called ...
References
* {{Citation, last=Hall, first=Brian C., title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, edition=2nd, series=Graduate Texts in Mathematics, volume=222, publisher=Springer, year=2015
External links
Are abelian non-degenerate tensor categories semisimple?
*http://ncatlab.org/nlab/show/semisimple+category
Linear algebra
Representation theory
Ring theory
Algebraic geometry