Representation theory is a branch ofmathematics that studiesabstractalgebraic structures byrepresenting theirelements aslinear transformations ofvector spaces, and studiesmodules over these abstract algebraic structures.[1][2] In essence, a representation makes an abstract algebraic object more concrete by describing its elements bymatrices and theiralgebraic operations (for example,matrix addition,matrix multiplication).
Thealgebraic objects amenable to such a description includegroups,associative algebras andLie algebras. The most prominent of these (and historically the first) is therepresentation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.[3][4]
Representation theory is a useful method because it reduces problems inabstract algebra to problems inlinear algebra, a subject that is well understood.[5][6] Representations of more abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories. For instance, representing a group by an infinite-dimensionalHilbert space allows methods ofanalysis to be applied to the theory of groups.[7][8] Furthermore, representation theory is important inphysics because it can describe how thesymmetry group of a physical system affects the solutions of equations describing that system.[9]
Representation theory is pervasive across fields of mathematics. The applications of representation theory are diverse.[10] In addition to its impact on algebra, representation theory
There are many approaches to representation theory: the same objects can be studied using methods fromalgebraic geometry,module theory,analytic number theory,differential geometry,operator theory,algebraic combinatorics andtopology.[14]
The success of representation theory has led to numerous generalizations. One of the most general is incategory theory.[15] The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations asfunctors from the object category to thecategory of vector spaces.[4] This description points to two natural 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.
Let be avector space over afield.[6] For instance, suppose is or, the standardn-dimensional space ofcolumn vectors over thereal orcomplex numbers, respectively. In this case, the idea of representation theory is to doabstract algebra concretely by usingmatrices of real or complex numbers.
There are three main sorts ofalgebraic objects for which this can be done:groups,associative algebras andLie algebras.[16][4]
This generalizes to any field and any vector space over, withlinear maps replacing matrices andcomposition replacing matrix multiplication: there isa group ofautomorphisms of, an associative algebra of all endomorphisms of, and a corresponding Lie algebra.
There are two ways to define a representation.[17] The first uses the idea of anaction, generalizing the way that matrices act on column vectors by matrix multiplication.
Arepresentation of agroup or (associative or Lie) algebra on a vector space is a mapwith two properties.
The definition for associative algebras is analogous, except that associative algebras do not always have an identity element, in which case equation (2.1) is omitted. Equation (2.2) 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 anyx1,x2 inA andv inV:where [x1,x2] is theLie bracket, which generalizes the matrix commutatorMN −NM.
The second way to define a representation focuses on the mapφ sendingg inG to a linear mapφ(g):V →V, which satisfies
and similarly in the other cases. This approach is both more concise and more abstract.From this point of view:
The vector spaceV is called therepresentation space ofφ and itsdimension (if finite) is called thedimension of the representation (sometimesdegree, as in[18]). It is also common practice to refer toV itself as the representation when the homomorphismφ is clear from the context; otherwise the notation (V,φ) can be used to denote a representation.
WhenV is of finite dimensionn, one can choose abasis forV to identifyV withFn, and hence recover a matrix representation with entries in the fieldF.
An effective orfaithful representation is a representation (V,φ), for which the homomorphismφ isinjective.
IfV andW are vector spaces overF, equipped with representationsφ andψ of a groupG, then anequivariant map fromV toW is a linear mapα:V →W such that
for allg inG andv inV. In terms ofφ:G → GL(V) andψ:G → GL(W), this means
for allg inG, that is, the followingdiagram commutes:
Equivariant maps for representations of an associative or Lie algebra are defined similarly. Ifα is invertible, then it is said to be anisomorphism, in which caseV andW (or, more precisely,φ andψ) areisomorphic representations, also phrased asequivalent representations. An equivariant map is often called anintertwining map of representations. Also, in the case of a groupG, it is on occasion called aG-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 representationsup to isomorphism.
If is a representation of (say) a group, and is a linear subspace of that is preserved by the action of in the sense that for all and, (Serre calls thesestable under[18]), then is called asubrepresentation: by defining where is the restriction of to, is a representation of and the inclusion of is an equivariant map. Thequotient space can also be made into a representation of. If has exactly two subrepresentations, namely thetrivial subspace {0} and itself, then the representation is said to beirreducible; if has a proper nontrivial subrepresentation, the representation is said to bereducible.[19]
The definition of an irreducible representation impliesSchur's lemma: an equivariant map between irreducible representations is either thezero map or an isomorphism, since itskernel andimage are subrepresentations. In particular, when, this shows that the equivariantendomorphisms of form an associativedivision algebra over the underlying fieldF. IfF isalgebraically 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 is not irreducible then it is built from a subrepresentation and a quotient that are both "simpler" in some sense; for instance, if 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 has a two dimensional representationThis group has the vector fixed by this homomorphism, but the complement subspace maps togiving only one irreducible subrepresentation. This is true for allunipotent groups.[20]: 112
If (V,φ) and (W,ψ) are representations of (say) a groupG, then thedirect sum ofV andW is a representation, in a canonical way, via the equation
Thedirect sum of two representations carries no more information about the groupG 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.
In favorable circumstances, every finite-dimensional representation is a direct sum of irreducible representations: such representations are said to besemisimple. In this case, it suffices to understand only the irreducible representations. Examples where this "complete reducibility" phenomenon occurs (at least over fields ofcharacteristic zero) include finite groups (seeMaschke's theorem), 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 by usingextensions of quotients by subrepresentations.
Suppose and are representations of a group. Then we can form a representation of G acting on thetensor product vector space as follows:[21]
If and are representations of a Lie algebra, then the correct formula to use is[22]
This product can be recognized as thecoproduct on acoalgebra. In general, the tensor product of irreducible representations isnot irreducible; the process of decomposing a tensor product as a direct sum of irreducible representations is known asClebsch–Gordan theory.
In the case of therepresentation theory of the group SU(2) (or equivalently, of its complexified Lie algebra), the decomposition is easy to work out.[23] The irreducible representations are labeled by a parameter that is a non-negative integer or half integer; the representation then has dimension. Suppose we take the tensor product of the representation of two representations, with labels and where we assume. Then the tensor product decomposes as a direct sum of one copy of each representation with label, where ranges from to in increments of 1. If, for example,, then the values of that occur are 0, 1, and 2. Thus, the tensor product representation of dimension decomposes as a direct sum of a 1-dimensional representation a 3-dimensional representation and a 5-dimensional representation.
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:
Group representations are a very important tool in the study of finite groups.[24] They also arise in the applications of finite group theory to geometry andcrystallography.[25] 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 ofcharacteristic zero, the representation of a finite groupG has a number of convenient properties. First, the representations ofG are semisimple (completely reducible). This is a consequence ofMaschke's theorem, which states that any subrepresentationV of aG-representationW has aG-invariant complement. One proof is to choose anyprojectionπ fromW toV and replace it by its averageπG defined by
πG is equivariant, and its kernel is the required complement.
The finite-dimensionalG-representations can be understood usingcharacter theory: the character of a representationφ:G → GL(V) is the class functionχφ:G →F defined by
where is thetrace. An irreducible representation ofG is completely determined by its character.
Maschke's theorem holds more generally for fields ofpositive characteristicp, such as thefinite fields, as long as the primep iscoprime to theorder ofG. Whenp and |G| have acommon factor, there areG-representations that are not semisimple, which are studied in a subbranch calledmodular representation theory.
Averaging techniques also show that ifF is the real or complex numbers, then anyG-representation preserves aninner product onV in the sense that
for allg inG andv,w inW. Hence anyG-representation isunitary.
Unitary representations are automatically semisimple, since Maschke's result can be proven by taking theorthogonal 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 forcompact topological groups (including compact Lie groups), usingHaar measure, and the resulting theory is known asabstract harmonic analysis.
Over arbitrary fields, another class of finite groups that have a good representation theory are thefinite groups of Lie type. Important examples arelinear algebraic groups over finite fields. The representation theory of linear algebraic groups andLie groups extends these examples to infinite-dimensional groups, the latter being intimately related toLie algebra representations. The importance of character theory for finite groups has an analogue in the theory ofweights for representations of Lie groups and Lie algebras.
Representations of a finite groupG are also linked directly to algebra representations via thegroup algebraF[G], which is a vector space overF with the elements ofG 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 of a finite groupG 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 inF and so one cannot divide by it).[26] 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, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because theirSylow 2-subgroups were "too small".[27]
As well as having applications to group theory, modular representations arise naturally in other branches ofmathematics, such asalgebraic geometry,coding theory,combinatorics andnumber theory.
A unitary representation of a groupG is a linear representationφ ofG on a real or (usually) complexHilbert spaceV such thatφ(g) is aunitary operator for everyg ∈G. Such representations have been widely applied inquantum mechanics since the 1920s, thanks in particular to the influence ofHermann Weyl,[28] and this has inspired the development of the theory, most notably through the analysis ofrepresentations of the Poincaré group byEugene Wigner.[29] One of the pioneers in constructing a general theory of unitary representations (for any groupG rather than just for particular groups useful in applications) wasGeorge Mackey, and an extensive theory was developed byHarish-Chandra and others in the 1950s and 1960s.[30]
A major goal is to describe the "unitary dual", the space of irreducible unitary representations ofG.[31] The theory is most well-developed in the case thatG is alocally compact (Hausdorff)topological group and the representations arestrongly continuous.[11] ForG abelian, the unitary dual is just the space ofcharacters, while forG compact, thePeter–Weyl theorem shows that the irreducible unitary representations are finite-dimensional and the unitary dual is discrete.[32] For example, ifG is the circle groupS1, then the characters are given by integers, and the unitary dual isZ.
For non-compactG, the question of which representations are unitary is a subtle one. Although irreducible unitary representations must be "admissible" (asHarish-Chandra modules) and it is easy to detect which admissible representations have a nondegenerate invariantsesquilinear 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 realreductiveLie groups (discussed below), remains an important open problem in representation theory. It has been solved for many particular groups, such asSL(2,R) and theLorentz group.[33]
The duality between the circle groupS1 and the integersZ, or more generally, between a torusTn andZn is well known in analysis as the theory ofFourier series, and theFourier transform similarly expresses the fact that the space of characters on a real vector space is thedual vector space. Thus unitary representation theory andharmonic analysis are intimately related, and abstract harmonic analysis exploits this relationship, by developing theanalysis of functions onlocally compact topological groups and related spaces.[11]
A major goal is to provide a general form of the Fourier transform and thePlancherel theorem. This is done by constructing ameasure on theunitary dual and an isomorphism between theregular representation ofG on the space L2(G) ofsquare integrable functions onG and its representation on thespace of L2 functions on the unitary dual.Pontrjagin duality and thePeter–Weyl theorem achieve this for abelian and compactG respectively.[32][34]
Another approach involves considering all unitary representations, not just the irreducible ones. These form acategory, andTannaka–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, althoughAlexander Grothendieck extended Tannaka–Krein duality to a relationship betweenlinear algebraic groups andtannakian categories.
Harmonic analysis has also been extended from the analysis of functions on a groupG to functions onhomogeneous spaces forG. The theory is particularly well developed forsymmetric spaces and provides a theory ofautomorphic forms (discussed below).
| Lie groups andLie algebras |
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ALie group is a group that is also asmooth manifold. Many classical groups of matrices over the real or complex numbers are Lie groups.[35] 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.[9]
The representation theory of Lie groups can be developed first by considering the compact groups, to which results of compact representation theory apply.[31] This theory can be extended to finite-dimensional representations ofsemisimple Lie groups usingWeyl's unitary trick: each semisimple real Lie groupG has a complexification, which is a complex Lie groupGc, and this complex Lie group has a maximal compact subgroupK. The finite-dimensional representations ofG closely correspond to those ofK.
A general Lie group is asemidirect product of asolvable Lie group and a semisimple Lie group (theLevi decomposition).[36] 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 calledMackey theory, which is a generalization of the methods used inWigner's classification of representations of the Poincaré group.
ALie algebra over a fieldF is a vector space overF equipped with askew-symmetricbilinear operation called theLie bracket, which satisfies theJacobi identity. Lie algebras arise in particular astangent spaces toLie groups at theidentity element, leading to their interpretation as "infinitesimal symmetries".[36] 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.[37]
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 ofÉlie Cartan. A representation of a semisimple Lie algebra 𝖌 is analysed by choosing aCartan 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 intoweight spaces that areeigenspaces 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.[36]
There are many classes of infinite-dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac–Moody algebras.[38] They are named afterVictor Kac andRobert Moody, who independently discovered them. These algebras form a generalization of finite-dimensionalsemisimple Lie algebras, 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 andtheoretical physics, especiallyconformal field theory and the theory ofexactly 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 are generalizations of Lie algebras in which the underlying vector space has aZ2-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.[39]
Linear algebraic groups (or more generally, affinegroup schemes) are analogues in algebraic geometry ofLie groups, but over more general fields than justR orC. In particular, over finite fields, they give rise tofinite 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 theZariski topology is relatively weak, and techniques from analysis are no longer available.[40]
Invariant theory studiesactions onalgebraic 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 ofpolynomial functions that do not change, or areinvariant, under the transformations from a givenlinear group. The modern approach analyses the decomposition of these representations into irreducibles.[41]
Invariant theory ofinfinite groups is inextricably linked with the development oflinear algebra, especially, the theories ofquadratic forms anddeterminants. Another subject with strong mutual influence isprojective geometry, where invariant theory can be used to organize the subject, and during the 1960s, new life was breathed into the subject byDavid Mumford in the form of hisgeometric invariant theory.[42]
The representation theory ofsemisimple Lie groups has its roots in invariant theory[35] and the strong links between representation theory and algebraic geometry have many parallels in differential geometry, beginning withFelix Klein'sErlangen program andÉlie Cartan'sconnections, which place groups and symmetry at the heart of geometry.[43] Modern developments link representation theory and invariant theory to areas as diverse asholonomy,differential operators and the theory ofseveral complex variables.
Automorphic forms are a generalization ofmodular forms to more generalanalytic functions, perhaps ofseveral complex variables, with similar transformation properties.[44] The generalization involves replacing the modular groupPSL2 (R) and a chosencongruence subgroup by a semisimple Lie groupG and adiscrete subgroupΓ. Just as modular forms can be viewed asdifferential forms on a quotient of theupper half spaceH = PSL2 (R)/SO(2), automorphic forms can be viewed as differential forms (or similar objects) onΓ\G/K, whereK is (typically) amaximal compact subgroup ofG. Some care is required, however, as the quotient typically has singularities. The quotient of a semisimple Lie group by a compact subgroup is asymmetric space 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 theHilbert modular forms andSiegel modular forms. Important results in the theory include theSelberg trace formula and the realization byRobert Langlands that theRiemann–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 thatG is analgebraic group, treated as anadelic algebraic group. As a result, an entire philosophy, theLanglands program has developed around the relation between representation and number theoretic properties of automorphic forms.[45]
In one sense,associative algebra representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a correspondinggroup ring orgroup algebra, while representations of a Lie algebra correspond bijectively to representations of itsuniversal 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.
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 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 asquantum groups, 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 thecrystal basis of Kashiwara.
Aset-theoretic representation (also known as agroup action orpermutation representation) of agroupG on asetX is given by afunctionρ fromG toXX, theset offunctions fromX toX, such that for allg1,g2 inG and allx inX:
This condition and the axioms for a group imply thatρ(g) is abijection (orpermutation) for allg inG. Thus we may equivalently define a permutation representation to be agroup homomorphism from G to thesymmetric group SX ofX.
Every groupG can be viewed as acategory with a single object;morphisms in this category are just the elements ofG. Given an arbitrary categoryC, arepresentation ofG inC is afunctor fromG toC. Such a functor selects an objectX inC and a group homomorphism fromG to Aut(X), theautomorphism group ofX.
In the case whereC isVectF, thecategory of vector spaces over a fieldF, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation ofG in thecategory of sets.
For another example consider thecategory of topological spaces,Top. Representations inTop are homomorphisms fromG to thehomeomorphism group of a topological spaceX.
Three types of representations closely related to linear representations are:
Since groups are categories, one can also consider representation of other categories. The simplest generalization is tomonoids, 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 aremonoid 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 offunctors between categories, and little can be said.
One special case has had a significant impact on representation theory, namely the representation theory of quivers.[15] A quiver is simply adirected graph (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.
For now, see the following.
linear algebra is very well understood
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