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 between
finite-dimensional 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,
Banach space, 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 fields, 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 and
crystallography. 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, 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
πG defined by : \pi_G(x) = \frac1\sum_{g\in G} g\cdot \pi(g^{-1}\cdot x).
πG is equivariant, and its kernel is the required complement. The finite-dimensional
G-representations can be understood using
character theory: the character of a representation
φ:
G → GL(
V) is the class function
χφ:
G →
F defined by :\chi_{\varphi}(g) = \mathrm{Tr}(\varphi(g)) where \mathrm{Tr} is the
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 fields, as long as the prime
p is
coprime to the
order of
G. When
p and |
G| have a
common factor, 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 \langle\cdot,\cdot\rangle on
V in the sense that :\langle g\cdot v,g\cdot w\rangle = \langle v,w\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, 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 groups over finite fields. The representation theory of linear algebraic groups and
Lie groups extends these examples to infinite-dimensional groups, the latter being intimately related to
Lie algebra representations. 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 the
classification of finite simple groups, 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, such as
algebraic geometry,
coding theory,
combinatorics and
number theory.
Unitary representations A unitary representation of a group
G is a linear representation
φ of
G on a real or (usually) complex
Hilbert space V such that
φ(
g) is a
unitary operator for every
g ∈
G. Such representations have been widely applied in
quantum mechanics since the 1920s, thanks in particular to the influence of
Hermann Weyl, and this has inspired the development of the theory, most notably through the analysis of
representations of the Poincaré group by
Eugene Wigner. 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, 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 and the representations are
strongly continuous. For example, if
G is the circle group
S1, 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 Lie groups (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
S1 and the integers
Z, or more generally, between a torus
Tn and
Zn is well known in analysis as the theory of
Fourier series, and the
Fourier transform 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 of functions on
locally compact topological groups and related spaces. Another approach involves considering all unitary representations, not just the irreducible ones. These form a
category, 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 groups 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 spaces and provides a theory of
automorphic forms (discussed below).
Lie groups A
Lie group is a group that is also a
smooth manifold. 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 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 A
Lie algebra 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. Lie algebras arise in particular as
tangent spaces to
Lie groups at the
identity element, leading to their interpretation as "infinitesimal symmetries". 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. They are named after
Victor Kac and
Robert Moody, who independently discovered them. These algebras form a generalization of finite-dimensional
semisimple 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 and
theoretical physics, 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
Z2-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.
Linear algebraic groups Linear algebraic groups (or more generally, affine
group schemes) are analogues in algebraic geometry of
Lie groups, 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
Zariski topology is relatively weak, and techniques from analysis are no longer available.
Invariant theory Invariant theory studies
actions 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 forms and
determinants. Another subject with strong mutual influence is
projective geometry, where invariant theory can be used to organize the subject, and during the 1960s, new life was breathed into the subject by
David Mumford in the form of his
geometric invariant theory. The representation theory of
semisimple Lie groups has its roots in invariant theory Modern developments link representation theory and invariant theory to areas as diverse as
holonomy,
differential operators and the theory of
several complex variables.
Automorphic forms and number theory Automorphic forms are a generalization of
modular forms to more general
analytic functions, perhaps of
several complex variables, with similar transformation properties. The generalization involves replacing the modular group
PSL2 (R) and a chosen
congruence subgroup by a semisimple Lie group
G and a
discrete subgroup . Just as modular forms can be viewed as
differential forms on a quotient of the
upper half space H = PSL2 (
R)/SO(2), automorphic forms can be viewed as differential forms (or similar objects) on , 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
symmetric 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 the
Hilbert modular forms and
Siegel modular forms. Important results in the theory include the
Selberg trace formula 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 representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a corresponding
group ring 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 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 as
quantum 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 the
crystal basis of Kashiwara. ==History==