Representation theory is a branch of

^{''n''} or C^{''n''}, the standard ''n''-dimensional space of _{F}(''V'') of all endomorphisms of ''V'', and a corresponding Lie algebra gl(''V'',F).

_{1}, ''g''_{2} in ''G'' and ''v'' in ''V'':
:$(1)\backslash quad\; e\; \backslash cdot\; v\; =\; v$
:$(2)\backslash quad\; g\_1\backslash cdot\; (g\_2\; \backslash cdot\; v)\; =\; (g\_1g\_2)\; \backslash cdot\; v$
where ''e'' is the _{1}''g''_{2} is the product in ''G''. The requirement for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (1) is ignored. Equation (2) is an abstract expression of the associativity of matrix multiplication. This doesn't hold for the matrix commutator and also there is no identity element for the commutator. Hence for Lie algebras, the only requirement is that for any ''x''_{1}, ''x''_{2} in ''A'' and ''v'' in ''V'':
:$(2\text{\'})\backslash quad\; x\_1\backslash cdot\; (x\_2\; \backslash cdot\; v)\; -\; x\_2\backslash cdot\; (x\_1\; \backslash cdot\; v)\; =;\; href="/html/ALL/s/\_1,x\_2.html"\; ;"title="\_1,x\_2">\_1,x\_2$
where 1, ''x''2">'x''_{1}, ''x''_{2}is the Lie bracket, which generalizes the matrix commutator ''MN'' − ''NM''.

_{F}(''V'');
* a representation of a Lie algebra 𝖆 on a vector space ''V'' is a Lie algebra homomorphism ''φ'': 𝖆 → gl(''V'',F).

^{''n''}, and hence recover a matrix representation with entries in the field F.
An effective or faithful representation is a representation (''V'',''φ''), for which the homomorphism ''φ'' is

_{''G''} defined by
:$\backslash pi\_G(x)\; =\; \backslash frac1\backslash sum\_\; g\backslash cdot\; \backslash pi(g^\backslash cdot\; x).$
''π''_{''G''} is equivariant, and its kernel is the required complement.
The finite-dimensional ''G''-representations can be understood using _{''φ''}: ''G'' → F defined by
:$\backslash chi\_(g)\; =\; \backslash mathrm(\backslash varphi(g))$
where $\backslash mathrm$ is the

^{1}, then the characters are given by integers, and the unitary dual is Z.
For non-compact ''G'', the question of which representations are unitary is a subtle one. Although irreducible unitary representations must be "admissible" (as Harish-Chandra modules) and it is easy to detect which admissible representations have a nondegenerate invariant sesquilinear form, it is hard to determine when this form is positive definite. An effective description of the unitary dual, even for relatively well-behaved groups such as real reductive

^{1} and the integers Z, or more generally, between a torus ''T''^{''n''} and Z^{''n''} is well known in analysis as the theory of ^{2}(''G'') of ^{2} functions on the unitary dual. Pontrjagin duality and the Peter–Weyl theorem achieve this for abelian and compact ''G'' respectively.
Another approach involves considering all unitary representations, not just the irreducible ones. These form a

^{c}, and this complex Lie group has a maximal compact subgroup ''K''. The finite-dimensional representations of ''G'' closely correspond to those of ''K''.
A general Lie group is a

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 additi ...

s at the

_{2}-grading, and skew-symmetry and Jacobi identity properties of the Lie bracket are modified by signs. Their representation theory is similar to the representation theory of Lie algebras.

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 additi ...

s, but over more general fields than just R or C. In particular, over finite fields, they give rise to finite groups of Lie type. Although linear algebraic groups have a classification that is very similar to that of Lie groups, their representation theory is rather different (and much less well understood) and requires different techniques, since the

_{2} (R) and a chosen congruence subgroup by a semisimple Lie group ''G'' and a discrete subgroup ''Γ''. Just as modular forms can be viewed as _{2} (R)/SO(2), automorphic forms can be viewed as differential forms (or similar objects) on ''Γ''\''G''/''K'', where ''K'' is (typically) a maximal compact subgroup of ''G''. Some care is required, however, as the quotient typically has singularities. The quotient of a semisimple Lie group by a compact subgroup is a

^{''X''}, the _{1}, ''g''_{2} in ''G'' and all ''x'' in ''X'':
:$\backslash rho(1);\; href="/html/ALL/s/.html"\; ;"title="">$
:$\backslash rho(g\_1\; g\_2);\; href="/html/ALL/s/.html"\; ;"title="">$
This_condition_and_the_axioms_for_a_group_imply_that_''ρ''(''g'')_is_a_bijection.html" ;"title=".html" ;"title="rho(g_2)[x">rho(g_2)[x.
This condition and the axioms for a group imply that ''ρ''(''g'') is a bijection">.html" ;"title="rho(g_2)[x">rho(g_2)[x.
This condition and the axioms for a group imply that ''ρ''(''g'') is a bijection (or permutation) for all ''g'' in ''G''. Thus we may equivalently define a permutation representation to be a _{''X''} of ''X''.

_{F}, the category of vector spaces over a field F, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of ''G'' in the

An introduction to Lie groups and Lie algebras

(2008). Textbook, preliminary version pdf downloadable from author's home page. * Kevin Hartnett

(2020), article on representation theory in Quanta magazine

{{bots, deny=Yobot

mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

that studies abstract algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...

s by ''representing'' their elements as linear transformation
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 pr ...

s of 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 ...

s, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operation
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...

s (for example, matrix addition
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. However, there are other operations which could also be considered addition for matrices, such as the direct sum and the Kron ...

, matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.
The algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...

ic objects amenable to such a description include groups, associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...

s and Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

s. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.
Representation theory is a useful method because it reduces problems in 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 ter ...

to problems in linear algebra, a subject that is well understood.There are many textbooks on vector spaces and linear algebra. For an advanced treatment, see . Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

, methods of analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...

can be applied to the theory of groups. Representation theory is also important in physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rela ...

because, for example, it describes how the symmetry group
In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

of a physical system affects the solutions of equations describing that system..
Representation theory is pervasive across fields of mathematics for two reasons. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory:
* illuminates and generalizes Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

via harmonic analysis,.
* is connected to geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

via invariant theory and the Erlangen program,
* has an impact in number theory via automorphic forms and the Langlands program.
Second, there are diverse approaches to representation theory. The same objects can be studied using methods from 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 ...

, module theory, analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diri ...

, differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multil ...

, operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...

, algebraic combinatorics and topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

.
The success of representation theory has led to numerous generalizations. One of the most general is in 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 ...

.. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

s from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.
Definitions and concepts

Let ''V'' be avector 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 ...

over a field F. For instance, suppose ''V'' is Rcolumn vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, ...

s over the real or 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, respectively. In this case, the idea of representation theory is to do 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 ter ...

concretely by using ''n'' × ''n'' matrices of real or complex numbers.
There are three main sorts of algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...

ic objects for which this can be done: groups, associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...

s and Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

s.
* The set of all '' invertible'' ''n'' × ''n'' matrices is a group under matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...

, and the representation theory of groups analyzes a group by describing ("representing") its elements in terms of invertible matrices.
* Matrix addition and multiplication make the set of ''all'' ''n'' × ''n'' matrices into an associative algebra, and hence there is a corresponding representation theory of associative algebras.
* If we replace matrix multiplication ''MN'' by the matrix commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.
Group theory
The commutator of two elements, a ...

''MN'' − ''NM'', then the ''n'' × ''n'' matrices become instead a Lie algebra, leading to a representation theory of Lie algebras.
This generalizes to any field F and any vector space ''V'' over F, with linear map
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 ...

s replacing matrices and composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
* Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include ...

replacing matrix multiplication: there is a group GL(''V'',F) of automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...

s of ''V'', an associative algebra EndDefinition

Action

There are two ways to say what a representation is. The first uses the idea of anaction
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...

, generalizing the way that matrices act on column vectors by matrix multiplication. A representation of a group ''G'' or (associative or Lie) algebra ''A'' on a vector space ''V'' is a map
:$\backslash Phi\backslash colon\; G\backslash times\; V\; \backslash to\; V\; \backslash quad\backslash text\backslash quad\; \backslash Phi\backslash colon\; A\backslash times\; V\; \backslash to\; V$
with two properties. First, for any ''g'' in ''G'' (or ''a'' in ''A''), the map
:$\backslash begin\backslash Phi(g)\backslash colon\; V\&\; \backslash to\; V\backslash \backslash \; v\; \&\; \backslash mapsto\; \backslash Phi(g,\; v)\backslash end$
is linear (over F). Second, if we introduce the notation ''g'' · ''v'' for $\backslash Phi$ (''g'', ''v''), then for any ''g''identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

of ''G'' and ''g''Mapping

The second way to define a representation focuses on the map ''φ'' sending ''g'' in ''G'' to a linear map ''φ''(''g''): ''V'' → ''V'', which satisfies :$\backslash varphi(g\_1\; g\_2)\; =\; \backslash varphi(g\_1)\backslash circ\; \backslash varphi(g\_2)\; \backslash quad\; \backslash textg\_1,g\_2\; \backslash in\; G$ and similarly in the other cases. This approach is both more concise and more abstract. From this point of view: * a representation of a group ''G'' on a vector space ''V'' is agroup homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
...

''φ'': ''G'' → GL(''V'',F);
* a representation of an associative algebra ''A'' on a vector space ''V'' is an algebra homomorphism ''φ'': ''A'' → EndTerminology

The vector space ''V'' is called the representation space of ''φ'' and itsdimension
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 ...

(if finite) is called the dimension of the representation (sometimes ''degree'', as in .). It is also common practice to refer to ''V'' itself as the representation when the homomorphism ''φ'' is clear from the context; otherwise the notation (''V'',''φ'') can be used to denote a representation.
When ''V'' is of finite dimension ''n'', one can choose a basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting o ...

for ''V'' to identify ''V'' with Finjective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...

.
Equivariant maps and isomorphisms

If ''V'' and ''W'' are vector spaces over F, equipped with representations ''φ'' and ''ψ'' of a group ''G'', then an equivariant map from ''V'' to ''W'' is a linear map ''α'': ''V'' → ''W'' such that :$\backslash alpha(\; g\backslash cdot\; v\; )\; =\; g\; \backslash cdot\; \backslash alpha(v)$ for all ''g'' in ''G'' and ''v'' in ''V''. In terms of ''φ'': ''G'' → GL(''V'') and ''ψ'': ''G'' → GL(''W''), this means :$\backslash alpha\backslash circ\; \backslash varphi(g)\; =\; \backslash psi(g)\backslash circ\; \backslash alpha$ for all ''g'' in ''G'', that is, the following diagram commutes: : Equivariant maps for representations of an associative or Lie algebra are defined similarly. If ''α'' is invertible, then it is said to be anisomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

, in which case ''V'' and ''W'' (or, more precisely, ''φ'' and ''ψ'') are ''isomorphic representations'', also phrased as ''equivalent representations''. An equivariant map is often called an ''intertwining map'' of representations. Also, in the case of a group , it is on occasion called a -map.
Isomorphic representations are, for practical purposes, "the same"; they provide the same information about the group or algebra being represented. Representation theory therefore seeks to classify representations up to isomorphism.
Subrepresentations, quotients, and irreducible representations

If $(V,\backslash psi)$ is a representation of (say) a group $G$, and $W$ is a linear subspace of $V$ that is preserved by the action of $G$ in the sense that for all $w\; \backslash in\; W$ and $g\backslash in\; G$, $g\backslash cdot\; w\; \backslash in\; W$ ( Serre calls these $W$ ''stable under'' $G$), then $W$ is called a '' subrepresentation'': by defining $$\backslash phi:G\; \backslash to\; \backslash text(W)$$ where $\backslash phi(g)$ is the restriction of $\backslash psi(g)$ to $W$, $(W,\backslash phi)$ is a representation of $G$ and the inclusion of $W\; \backslash hookrightarrow\; V$ is an equivariant map. The quotient space $V/W$ can also be made into a representation of $G$. If $V$ has exactly two subrepresentations, namely the trivial subspace and $V$ itself, then the representation is said to be ''irreducible''; if $V$ has a proper nontrivial subrepresentation, the representation is said to be ''reducible''. The definition of an irreducible representation impliesSchur'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 ...

: an equivariant map $$\backslash alpha:\; (V,\backslash psi)\; \backslash to\; (V\text{'},\backslash psi\text{'})$$ between irreducible representations is either the zero map or an isomorphism, since its kernel and image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimension ...

are subrepresentations. In particular, when $V\; =\; V\text{'}$, this shows that the equivariant 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 gr ...

s of $V$ form an associative division algebra over the underlying field F. If F is algebraically closed, the only equivariant endomorphisms of an irreducible representation are the scalar multiples of the identity.
Irreducible representations are the building blocks of representation theory for many groups: if a representation $V$ is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if $V$ is finite-dimensional, then both the subrepresentation and the quotient have smaller dimension. There are counterexamples where a representation has a subrepresentation, but only has one non-trivial irreducible component. For example, the additive group $(\backslash mathbb,+)$ has a two dimensional representation
$$\backslash phi(a)\; =\; \backslash begin\; 1\; \&\; a\; \backslash \backslash \; 0\; \&\; 1\; \backslash end$$
This group has the vector $\backslash begin\; 1\; \&\; 0\; \backslash end^\backslash mathsf$ fixed by this homomorphism, but the complement subspace maps to
$$\backslash begin\; 0\; \backslash \backslash \; 1\; \backslash end\; \backslash mapsto\; \backslash begin\; a\; \backslash \backslash \; 1\; \backslash end$$
giving only one irreducible subrepresentation. This is true for all unipotent groups.
Direct sums and indecomposable representations

If (''V'',''φ'') and (''W'',''ψ'') are representations of (say) a group ''G'', then the direct sum of ''V'' and ''W'' is a representation, in a canonical way, via the equation :$g\backslash cdot\; (v,w)\; =\; (g\backslash cdot\; v,\; g\backslash cdot\; w).$ The direct sum of two representations carries no more information about the group ''G'' than the two representations do individually. If a representation is the direct sum of two proper nontrivial subrepresentations, it is said to be decomposable. Otherwise, it is said to be indecomposable.Complete reducibility

In favorable circumstances, every finite-dimensional representation is a direct sum of irreducible representations: such representations are said to besemisimple
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 ' ...

. In this case, it suffices to understand only the irreducible representations. Examples where this " complete reducibility" phenomenon occur include finite groups (see 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 ...

), compact groups, and semisimple Lie algebras.
In cases where complete reducibility does not hold, one must understand how indecomposable representations can be built from irreducible representations as extensions of a quotient by a subrepresentation.
Tensor products of representations

Suppose $\backslash phi\_1:G\backslash rightarrow\; \backslash mathrm(V\_1)$ and $\backslash phi\_2:G\backslash rightarrow\; \backslash mathrm(V\_2)$ are representations of a group $G$. Then we can form a representation $\backslash phi\_1\backslash otimes\backslash phi\_2$ of G acting on thetensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes ...

vector space $V\_1\backslash otimes\; V\_2$ as follows:
:$(\backslash phi\_1\backslash otimes\backslash phi\_2)(g)=\backslash phi\_1(g)\backslash otimes\backslash phi\_2(g)$.
If $\backslash phi\_1$ and $\backslash phi\_2$ are representations of a Lie algebra, then the correct formula to use is
:$(\backslash phi\_1\backslash otimes\backslash phi\_2)(X)=\backslash phi\_1(X)\backslash otimes\; I+I\backslash otimes\backslash phi\_2(X)$.
This product can be recognized as the 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 copr ...

on a coalgebra In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagram ...

. In general, the tensor product of irreducible representations is ''not'' irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known as Clebsch–Gordan theory.
In the case of the representation theory of the group SU(2) (or equivalently, of its complexified Lie algebra $\backslash mathrm(2;\backslash mathbb)$), the decomposition is easy to work out. The irreducible representations are labeled by a parameter $l$ that is a non-negative integer or half integer; the representation then has dimension $2l+1$. Suppose we take the tensor product of the representation of two representations, with labels $l\_1$ and $l\_2,$ where we assume $l\_1\backslash geq\; l\_2$. Then the tensor product decomposes as a direct sum of one copy of each representation with label $l$, where $l$ ranges from $l\_1-l\_2$ to $l\_1+l\_2$ in increments of 1. If, for example, $l\_1=l\_2=1$, then the values of $l$ that occur are 0, 1, and 2. Thus, the tensor product representation of dimension $(2l\_1+1)\; \backslash times\; (2l\_2+1)\; =\; 3\backslash times\; 3=9$ decomposes as a direct sum of a 1-dimensional representation $(l=0),$ a 3-dimensional representation $(l=1),$ and a 5-dimensional representation $(l=2)$.
Branches and topics

Representation theory is notable for the number of branches it has, and the diversity of the approaches to studying representations of groups and algebras. Although, all the theories have in common the basic concepts discussed already, they differ considerably in detail. The differences are at least 3-fold: # Representation theory depends upon the type of algebraic object being represented. There are several different classes of groups, associative algebras and Lie algebras, and their representation theories all have an individual flavour. # Representation theory depends upon the nature of the vector space on which the algebraic object is represented. The most important distinction is betweenfinite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to dis ...

representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (for example, whether or not the space is a Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

, Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vecto ...

, etc.). Additional algebraic structures can also be imposed in the finite-dimensional case.
# Representation theory depends upon the type of field over which the vector space is defined. The most important cases are the field of complex numbers, the field of real numbers, finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

s, and fields of p-adic numbers. Additional difficulties arise for fields of positive characteristic and for fields that are not algebraically closed.
Finite groups

Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to geometry andcrystallography
Crystallography is the experimental science of determining the arrangement of atoms in crystalline solids. Crystallography is a fundamental subject in the fields of materials science and solid-state physics (condensed matter physics). The wor ...

. Representations of finite groups exhibit many of the features of the general theory and point the way to other branches and topics in representation theory.
Over a field of characteristic zero, the representation of a finite group ''G'' has a number of convenient properties. First, the representations of ''G'' are semisimple (completely reducible). This is a consequence of 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 ...

, which states that any subrepresentation ''V'' of a ''G''-representation ''W'' has a ''G''-invariant complement. One proof is to choose any projection ''π'' from ''W'' to ''V'' and replace it by its average ''π''character theory
In mathematics, more specifically in group theory, the character of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information abo ...

: the character of a representation ''φ'': ''G'' → GL(''V'') is the class function ''χ''trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace' ...

. An irreducible representation of ''G'' is completely determined by its character.
Maschke's theorem holds more generally for fields of positive characteristic ''p'', such as the finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...

s, as long as the prime ''p'' is coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equival ...

to the order of ''G''. When ''p'' and , ''G'', have a common factor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

, there are ''G''-representations that are not semisimple, which are studied in a subbranch called modular representation theory.
Averaging techniques also show that if F is the real or complex numbers, then any ''G''-representation preserves an inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...

$\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ on ''V'' in the sense that
:$\backslash langle\; g\backslash cdot\; v,g\backslash cdot\; w\backslash rangle\; =\; \backslash langle\; v,w\backslash rangle$
for all ''g'' in ''G'' and ''v'', ''w'' in ''W''. Hence any ''G''-representation is unitary.
Unitary representations are automatically semisimple, since Maschke's result can be proven by taking the orthogonal complement of a subrepresentation. When studying representations of groups that are not finite, the unitary representations provide a good generalization of the real and complex representations of a finite group.
Results such as Maschke's theorem and the unitary property that rely on averaging can be generalized to more general groups by replacing the average with an integral, provided that a suitable notion of integral can be defined. This can be done for compact topological groups (including compact Lie groups), using Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups.
This measure was introduced by Alfréd Haar in 1933, thou ...

, and the resulting theory is known as abstract harmonic analysis.
Over arbitrary fields, another class of finite groups that have a good representation theory are the finite groups of Lie type. Important examples are linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...

s over finite fields. The representation theory of linear algebraic groups and 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 additi ...

s extends these examples to infinite-dimensional groups, the latter being intimately related to Lie algebra representation
In the mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms of a vector space) in such a way that the Lie bracket is ...

s. The importance of character theory for finite groups has an analogue in the theory of weights for representations of Lie groups and Lie algebras.
Representations of a finite group ''G'' are also linked directly to algebra representations via the group algebra F 'G'' which is a vector space over F with the elements of ''G'' as a basis, equipped with the multiplication operation defined by the group operation, linearity, and the requirement that the group operation and scalar multiplication commute.
Modular representations

Modular representations of a finite group ''G'' are representations over a field whose characteristic is not coprime to , ''G'', , so that Maschke's theorem no longer holds (because , ''G'', is not invertible in F and so one cannot divide by it). Nevertheless, Richard Brauer extended much of character theory to modular representations, and this theory played an important role in early progress towards theclassification of finite simple groups
In mathematics, the classification of the finite simple groups is a result of group theory stating that every finite simple group is either cyclic, or alternating, or it belongs to a broad infinite class called the groups of Lie type, or else ...

, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2-subgroups were "too small".
As well as having applications to group theory, modular representations arise naturally in other branches of mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, such as 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 ...

, coding theory, combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many app ...

and number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...

.
Unitary representations

A unitary representation of a group ''G'' is a linear representation ''φ'' of ''G'' on a real or (usually) complexHilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...

''V'' such that ''φ''(''g'') is a unitary operator for every ''g'' ∈ ''G''. Such representations have been widely applied in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...

since the 1920s, thanks in particular to the influence of Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...

, and this has inspired the development of the theory, most notably through the analysis of representations of the Poincaré group by Eugene Wigner
Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his con ...

. One of the pioneers in constructing a general theory of unitary representations (for any group ''G'' rather than just for particular groups useful in applications) was George Mackey
George Whitelaw Mackey (February 1, 1916 – March 15, 2006) was an American mathematician known for his contributions to quantum logic, representation theory, and noncommutative geometry.
Career
Mackey earned his bachelor of arts at Rice Unive ...

, and an extensive theory was developed by Harish-Chandra and others in the 1950s and 1960s.
A major goal is to describe the " unitary dual", the space of irreducible unitary representations of ''G''.. The theory is most well-developed in the case that ''G'' is a locally compact (Hausdorff) topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two s ...

and the representations are strongly continuous. For ''G'' abelian, the unitary dual is just the space of characters, while for ''G'' compact, the Peter–Weyl theorem shows that the irreducible unitary representations are finite-dimensional and the unitary dual is discrete.. For example, if ''G'' is the circle group ''S''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 additi ...

s (discussed below), remains an important open problem in representation theory. It has been solved for many particular groups, such as SL(2,R) and the Lorentz group.
Harmonic analysis

The duality between the circle group ''S''Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...

, and the Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

similarly expresses the fact that the space of characters on a real vector space is the dual vector space. Thus unitary representation theory and harmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing the analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384 ...

of functions on locally compact topological groups and related spaces.
A major goal is to provide a general form of the Fourier transform and the Plancherel theorem
In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the inte ...

. This is done by constructing a measure on the unitary dual and an isomorphism between the regular representation of ''G'' on the space Lsquare integrable
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value ...

functions on ''G'' and its representation on the space of Lcategory
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)
...

, and Tannaka–Krein duality provides a way to recover a compact group from its category of unitary representations.
If the group is neither abelian nor compact, no general theory is known with an analogue of the Plancherel theorem or Fourier inversion, although Alexander Grothendieck extended Tannaka–Krein duality to a relationship between linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...

s and tannakian categories.
Harmonic analysis has also been extended from the analysis of functions on a group ''G'' to functions on homogeneous spaces for ''G''. The theory is particularly well developed for symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...

s and provides a theory of automorphic forms (discussed below).
Lie groups

ALie 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 additi ...

is a group that is also a smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...

. Many classical groups of matrices over the real or complex numbers are Lie groups.. Many of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields.
The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply. This theory can be extended to finite-dimensional representations of semisimple Lie groups using Weyl's unitary trick: each semisimple real Lie group ''G'' has a complexification, which is a complex Lie group ''G''semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:
* an ''inner'' semidirect product is a particular way in wh ...

of a solvable Lie group
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra. The ''derived Lie algebra'' of the Lie algebra \mathfrak is the subalgebra of \mathfrak, denoted
: mathfrak,\mathfrak/math>
that consist ...

and a semisimple Lie group (the Levi decomposition).. The classification of representations of solvable Lie groups is intractable in general, but often easy in practical cases. Representations of semidirect products can then be analysed by means of general results called '' Mackey theory'', which is a generalization of the methods used in Wigner's classification of representations of the Poincaré group.
Lie algebras

ALie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

over a field F is a vector space over F equipped with a skew-symmetric bilinear operation called the Lie bracket, which satisfies the Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...

. Lie algebras arise in particular as tangent spaces to identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...

, leading to their interpretation as "infinitesimal symmetries". An important approach to the representation theory of Lie groups is to study the corresponding representation theory of Lie algebras, but representations of Lie algebras also have an intrinsic interest.
Lie algebras, like Lie groups, have a Levi decomposition into semisimple and solvable parts, with the representation theory of solvable Lie algebras being intractable in general. In contrast, the finite-dimensional representations of semisimple Lie algebras are completely understood, after work of Élie Cartan. A representation of a semisimple Lie algebra 𝖌 is analysed by choosing a Cartan subalgebra, which is essentially a generic maximal subalgebra 𝖍 of 𝖌 on which the Lie bracket is zero ("abelian"). The representation of 𝖌 can be decomposed into weight spaces that are eigenspaces for the action of 𝖍 and the infinitesimal analogue of characters. The structure of semisimple Lie algebras then reduces the analysis of representations to easily understood combinatorics of the possible weights that can occur.
Infinite-dimensional Lie algebras

There are many classes of infinite-dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac–Moody algebras. They are named after Victor Kac and Robert Moody, who independently discovered them. These algebras form a generalization of finite-dimensionalsemisimple 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 is ...

s, and share many of their combinatorial properties. This means that they have a class of representations that can be understood in the same way as representations of semisimple Lie algebras.
Affine Lie algebras are a special case of Kac–Moody algebras, which have particular importance in mathematics and theoretical physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experime ...

, especially conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras.
Lie superalgebras

Lie superalgebras are generalizations of Lie algebras in which the underlying vector space has a ZLinear algebraic groups

Linear algebraic groups (or more generally, affinegroup scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...

s) are analogues in algebraic geometry of Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...

is relatively weak, and techniques from analysis are no longer available.
Invariant theory

Invariant theory studiesactions
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...

on algebraic varieties from the point of view of their effect on functions, which form representations of the group. Classically, the theory dealt with the question of explicit description of polynomial functions that do not change, or are ''invariant'', under the transformations from a given linear group. The modern approach analyses the decomposition of these representations into irreducibles.
Invariant theory of infinite groups is inextricably linked with the development of linear algebra, especially, the theories of quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...

s and determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...

s. Another subject with strong mutual influence is projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...

, where invariant theory can be used to organize the subject, and during the 1960s, new life was breathed into the subject by David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded ...

in the form of his geometric invariant theory.
The representation theory of semisimple Lie groups has its roots in invariant theory and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning with Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group ...

's Erlangen program and Élie Cartan's connections
Connections may refer to:
Television
* '' Connections: An Investigation into Organized Crime in Canada'', a documentary television series
* ''Connections'' (British documentary), a documentary television series and book by science historian Jam ...

, which place groups and symmetry at the heart of geometry. Modern developments link representation theory and invariant theory to areas as diverse as holonomy, differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

s and the theory of several complex variables.
Automorphic forms and number theory

Automorphic forms are a generalization ofmodular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory o ...

s to more general analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex a ...

s, perhaps of several complex variables, with similar transformation properties. The generalization involves replacing the modular group PSLdifferential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

s on a quotient of the upper half space ''H'' = PSLsymmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...

and so the theory of automorphic forms is intimately related to harmonic analysis on symmetric spaces.
Before the development of the general theory, many important special cases were worked out in detail, including the Hilbert modular forms and Siegel modular forms. Important results in the theory include the Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given ...

and the realization by Robert Langlands that the Riemann–Roch theorem could be applied to calculate the dimension of the space of automorphic forms. The subsequent notion of "automorphic representation" has proved of great technical value for dealing with the case that ''G'' is an algebraic group, treated as an adelic algebraic group. As a result, an entire philosophy, the Langlands program has developed around the relation between representation and number theoretic properties of automorphic forms..
Associative algebras

In one sense,associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplica ...

representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a corresponding 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 giv ...

or group algebra, while representations of a Lie algebra correspond bijectively to representations of its universal enveloping algebra. However, the representation theory of general associative algebras does not have all of the nice properties of the representation theory of groups and Lie algebras.
Module theory

When considering representations of an associative algebra, one can forget the underlying field, and simply regard the associative algebra as a ring, and its representations as modules. This approach is surprisingly fruitful: many results in representation theory can be interpreted as special cases of results about modules over a ring.Hopf algebras and quantum groups

Hopf algebra Hopf is a German surname. Notable people with the surname include:
*Eberhard Hopf (1902–1983), Austrian mathematician
*Hans Hopf (1916–1993), German tenor
*Heinz Hopf (1894–1971), German mathematician
*Heinz Hopf (actor) (1934–2001), Swedis ...

s provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and Lie algebras as special cases. In particular, the tensor product of two representations is a representation, as is the dual vector space.
The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopf algebras are known as quantum group
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...

s, although this term is often restricted to certain Hopf algebras arising as deformations of groups or their universal enveloping algebras. The representation theory of quantum groups has added surprising insights to the representation theory of Lie groups and Lie algebras, for instance through the crystal basis of Kashiwara.
Generalizations

Set-theoretic representations

A ''set-theoretic representation'' (also known as agroup action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

or ''permutation representation'') of a group ''G'' on a set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

''X'' is given by a function ''ρ'' from ''G'' to ''X''set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...

of functions from ''X'' to ''X'', such that for all ''g''group homomorphism
In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
: h(u*v) = h(u) \cdot h(v)
...

from G to the symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...

SRepresentations in other categories

Every group ''G'' can be viewed as acategory
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)
...

with a single object; morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

s in this category are just the elements of ''G''. Given an arbitrary category ''C'', a ''representation'' of ''G'' in ''C'' is a functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

from ''G'' to ''C''. Such a functor selects an object ''X'' in ''C'' and a group homomorphism from ''G'' to Aut(''X''), the automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the gr ...

of ''X''.
In the case where ''C'' is Vectcategory of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition o ...

.
For another example consider the category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...

, Top. Representations in Top are homomorphisms from ''G'' to the homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...

group of a topological space ''X''.
Three types of representations closely related to linear representations are:
* projective representations: in the category of projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

s. These can be described as "linear representations up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...

scalar transformations".
* affine representations: in the category of affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...

s. For example, the Euclidean group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations ...

acts affinely upon Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

.
* corepresentations of unitary and antiunitary groups: in the category of complex vector spaces with morphisms being linear or antilinear
In mathematics, a function f : V \to W between two complex vector spaces is said to be antilinear or conjugate-linear if
\begin
f(x + y) &= f(x) + f(y) && \qquad \text \\
f(s x) &= \overline f(x) && \qquad \text \\
\end
hold for all vectors x, y ...

transformations.
Representations of categories

Since groups are categories, one can also consider representation of other categories. The simplest generalization is tomonoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ar ...

s, which are categories with one object. Groups are monoids for which every morphism is invertible. General monoids have representations in any category. In the category of sets, these are monoid actions, but monoid representations on vector spaces and other objects can be studied.
More generally, one can relax the assumption that the category being represented has only one object. In full generality, this is simply the theory of functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

s between categories, and little can be said.
One special case has had a significant impact on representation theory, namely the representation theory of quivers. A quiver is simply a directed graph
In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.
Definition
In formal terms, a directed graph is an ordered pa ...

(with loops and multiple arrows allowed), but it can be made into a category (and also an algebra) by considering paths in the graph. Representations of such categories/algebras have illuminated several aspects of representation theory, for instance by allowing non-semisimple representation theory questions about a group to be reduced in some cases to semisimple representation theory questions about a quiver.
See also

* Galois representation * Glossary of representation theory *Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to r ...

* Itô's theorem
* List of representation theory topics
* List of harmonic analysis topics
* 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 distinguished from discrete mathematics). It is the study of numerical methods t ...

* Philosophy of cusp forms
* Representation (mathematics)
In mathematics, a representation is a very general relationship that expresses similarities (or equivalences) between mathematical objects or structures. Roughly speaking, a collection ''Y'' of mathematical objects may be said to ''represent'' anot ...

* Representation theorem
* Universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...

Notes

References

* . * . * . * . * . * . * . * . * . * . * * * . * * . * . * . * . * . * . * . * Yurii I. Lyubich. ''Introduction to the Theory of Banach Representations of Groups''. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988. *; (2nd ed.); (3rd ed.) * . * . * . * . * . * . * . * . * * . * . * .External links

* * Alexander Kirillov Jr.An introduction to Lie groups and Lie algebras

(2008). Textbook, preliminary version pdf downloadable from author's home page. * Kevin Hartnett

(2020), article on representation theory in Quanta magazine

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