In
mathematics, a compact (topological) group is a
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 st ...
whose
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 ...
realizes it as a
compact topological space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i. ...
(when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of
finite groups with the
discrete topology
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''isolated'' from each other in a certain sense. The discrete topology is the finest top ...
and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to
group 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 ...
s and
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
.
In the following we will assume all groups are
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s.
Compact Lie groups
Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include
* the
circle group T and the
torus group
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
s T
''n'',
* the
orthogonal group O(''n''), the
special orthogonal group SO(''n'') and its covering
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
Spin(''n''),
* the
unitary group
In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
U(''n'') and the
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
SU(''n''),
* the compact forms of the
exceptional Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
s:
G2,
F4,
E6,
E7, and
E8.
The
classification theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class.
A few issues relate ...
of compact Lie groups states that up to finite
extensions
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
and finite
covers this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.
Classification
Given any compact Lie group ''G'' one can take its
identity component
In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element.
In point set topology, the identity compo ...
''G''
0, which is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
. The
quotient group ''G''/''G''
0 is the group of components π
0(''G'') which must be finite since ''G'' is compact. We therefore have a finite extension
:
Meanwhile, for connected compact Lie groups, we have the following result:
:Theorem: Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.
Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.)
Finally, every compact, connected, simply-connected Lie group ''K'' is a product of finitely many compact, connected, simply-connected
simple Lie group
In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
s ''K''
''i'' each of which is isomorphic to exactly one of the following:
*The
compact symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gro ...
*The
special unitary group
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
*The
spin group
In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when )
:1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1.
As a ...
or one of the five exceptional groups
G2,
F4,
E6,
E7, and
E8. The restrictions on ''n'' are to avoid special isomorphisms among the various families for small values of ''n''. For each of these groups, the center is known explicitly. The classification is through the associated
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
(for a fixed maximal torus), which in turn are classified by their
Dynkin diagram
In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...
s.
The classification of compact, simply connected Lie groups is the same as the classification of complex
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s. Indeed, if ''K'' is a simply connected compact Lie group, then the complexification of the Lie algebra of ''K'' is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact, simply connected Lie group.
Maximal tori and root systems
A key idea in the study of a connected compact Lie group ''K'' is the concept of a ''maximal torus'', that is a subgroup ''T'' of ''K'' that is isomorphic to a product of several copies of
and that is not contained in any larger subgroup of this type. A basic example is the case
, in which case we may take
to be the group of diagonal elements in
. A basic result is the ''torus theorem'' which states that every element of
belongs to a maximal torus and that all maximal tori are conjugate.
The maximal torus in a compact group plays a role analogous to that of the
Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...
in a complex semisimple Lie algebra. In particular, once a maximal torus
has been chosen, one can define a
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
and a
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...
similar to what one has for
semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s. These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below).
The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:
*The special unitary groups
correspond to the root system
*The odd spin groups
correspond to the root system
*The compact symplectic groups
correspond to the root system
*The even spin groups
correspond to the root system
*The exceptional compact Lie groups correspond to the five exceptional root systems G
2, F
4, E
6, E
7, or E
8
Fundamental group and center
It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its
fundamental group. For compact Lie groups, there are
two basic approaches to computing the fundamental group. The first approach applies to the classical compact groups
,
,
, and
and proceeds by induction on
. The second approach uses the root system and applies to all connected compact Lie groups.
It is also important to know the center of a connected compact Lie group. The center of a classical group
can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in
. (The group SO(2) is an exception—the center is the whole group, even though most elements are not roots of the identity.) Thus, for example, the center of
consists of ''n''th roots of unity times the identity, a cyclic group of order
.
In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus. The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system
has trivial center. Thus,
the compact group is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are
and
.)
Further examples
Amongst groups that are not Lie groups, and so do not carry the structure of a
manifold, examples are the additive group ''Z''
''p'' of
p-adic integer
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...
s, and constructions from it. In fact any
profinite group is a compact group. This means that
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
s are compact groups, a basic fact for the theory of
algebraic extension
In mathematics, an algebraic extension is a field extension such that every element of the larger field is algebraic over the smaller field ; that is, if every element of is a root of a non-zero polynomial with coefficients in . A field ext ...
s in the case of infinite degree.
Pontryagin duality
In mathematics, Pontryagin duality is a duality (mathematics), duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numb ...
provides a large supply of examples of compact commutative groups. These are in duality with abelian
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
s.
Haar measure
Compact groups all carry a
Haar measure, which will be invariant by both left and right translation (the
modulus function must be a continuous
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
to
positive reals
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
(R
+, ×), and so 1). In other words, these groups are
unimodular. Haar measure is easily normalized to be a
probability measure, analogous to dθ/2π on the circle.
Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to
Adolf Hurwitz, and in the Lie group cases can always be given by an invariant
differential form. In the profinite case there are many subgroups of
finite index In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the
number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''.
The index is denoted , G:H, or :H/math> or (G ...
, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in
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, "Mat ...
.
If
is a compact group and
is the associated Haar measure, the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
provides a decomposition of
as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of
.
Representation theory
The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) was founded by the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
.
Hermann Weyl went on to give the detailed
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 ab ...
of the compact connected Lie groups, based on
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
theory. The resulting
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
was one of the influential results of twentieth century mathematics. The combination of the Peter–Weyl theorem and the Weyl character formula led Weyl to a complete classification of the representations of a connected compact Lie group; this theory is described in the next section.
A combination of Weyl's work and
Cartan's theorem gives a survey of the whole representation theory of compact groups ''G''. That is, by the Peter–Weyl theorem the irreducible
unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...
s ρ of ''G'' are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If ''G'' is not itself a Lie group, there must be a kernel to ρ. Further one can form an
inverse system
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits ca ...
, for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies ''G'' as an
inverse limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can ...
of compact Lie groups. Here the fact that in the limit a
faithful representation In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group on a vector space is a linear representation in which different elements of are represented by distinct linear map ...
of ''G'' is found is another consequence of the Peter–Weyl theorem.
The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the
complex representations of finite groups
The representation theory of groups is a part of mathematics which examines how groups act on given structures.
Here the focus is in particular on operations of groups on vector spaces. Nevertheless, groups acting on other groups or on sets are ...
. This theory is rather rich in detail, but is qualitatively well understood.
Representation theory of a connected compact Lie group
Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the
rotation group SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a ...
, the
special unitary group SU(2), and the
special unitary group SU(3). We focus here on the general theory. See also the parallel theory of
representations of a semisimple Lie algebra.
Throughout this section, we fix a connected compact Lie group ''K'' and a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
''T'' in ''K''.
Representation theory of ''T''
Since ''T'' is commutative,
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if ''M'' and ''N'' are two finite-dimensional irreducible representations
of a group ' ...
tells us that each irreducible representation
of ''T'' is one-dimensional:
:
Since, also, ''T'' is compact,
must actually map into
.
To describe these representations concretely, we let
be the Lie algebra of ''T'' and we write points
as
:
In such coordinates,
will have the form
:
for some linear functional
on
.
Now, since the exponential map
is not injective, not every such linear functional
gives rise to a well-defined map of ''T'' into
. Rather, let
denote the kernel of the exponential map:
:
where
is the identity element of ''T''. (We scale the exponential map here by a factor of
in order to avoid such factors elsewhere.)
Then for
to give a well-defined map
,
must satisfy
:
where
is the set of integers. A linear functional
satisfying this condition is called an analytically integral element. This integrality condition is related to, but not identical to, the notion of
integral element In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
in the setting of semisimple Lie algebras.
Suppose, for example, ''T'' is just the group
of complex numbers
of absolute value 1. The Lie algebra is the set of purely imaginary numbers,
and the kernel of the (scaled) exponential map is the set of numbers of the form
where
is an integer. A linear functional
takes integer values on all such numbers if and only if it is of the form
for some integer
. The irreducible representations of ''T'' in this case are one-dimensional and of the form
:
Representation theory of ''K''
![Weights_for_A2_root_system](https://upload.wikimedia.org/wikipedia/commons/1/11/Weights_for_A2_root_system.png)
We now let
denote a finite-dimensional irreducible representation of ''K'' (over
). We then consider the restriction of
to ''T''. This restriction is not irreducible unless
is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of ''T''. (Note that a given irreducible representation of ''T'' may occur more than once.) Now, each irreducible representation of ''T'' is described by a linear functional
as in the preceding subsection. If a given
occurs at least once in the decomposition of the restriction of
to ''T'', we call
a weight of
. The strategy of the representation theory of ''K'' is to classify the irreducible representations in terms of their weights.
We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on
weights in representation theory. We need the notion of a root system for ''K'' (relative to a given maximal torus ''T''). The construction of this root system
is very similar to the
construction for complex semisimple Lie algebras. Specifically, the weights are the nonzero weights for the adjoint action of ''T'' on the complexified Lie algebra of ''K''. The root system ''R'' has all the usual properties of a
root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representatio ...
, except that the elements of ''R'' may not span
. We then choose a base
for ''R'' and we say that an integral element
is dominant if
for all
. Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of
with non-negative coefficients.
The irreducible finite-dimensional representations of ''K'' are then classified by a theorem of the highest
weight
In science and engineering, the weight of an object is the force acting on the object due to gravity.
Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weigh ...
, which is closely related to the analogous theorem classifying
representations of a semisimple Lie algebra. The result says that:
# every irreducible representation has highest weight,
# the highest weight is always a dominant, analytically integral element,
# two irreducible representations with the same highest weight are isomorphic, and
# every dominant, analytically integral element arises as the highest weight of an irreducible representation.
The theorem of the highest weight for representations of ''K'' is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an
integral element In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that
:b^n + a_ b^ + \cdots + a_1 b + a_0 = 0.
That is to say, ''b'' is ...
is different. The weights
of a representation
are analytically integral in the sense described in the previous subsection. Every analytically integral element is
integral
In 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 i ...
in the Lie algebra sense, but not the other way around. (This phenomenon reflects that, in general,
not every representation of the Lie algebra
comes from a representation of the group ''K''.) On the other hand, if ''K'' is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.
The Weyl character formula
If
is representation of ''K'', we define the character of
to be the function
given by
:
.
This function is easily seen to be a class function, i.e.,
for all
and
in ''K''. Thus,
is determined by its restriction to ''T''.
The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
, is that the characters form an orthonormal basis for the set of square-integrable class functions in ''K''. A second key result is the
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
, which gives an explicit formula for the character—or, rather, the restriction of the character to ''T''—in terms of the highest weight of the representation.
In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established ''after'' the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of ''K'', the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in a totally different way from the usual Lie algebra construction using
Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.
Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Spe ...
s. In Weyl's approach, the construction is based on the
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
and an analytic proof of the
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
. Ultimately, the irreducible representations of ''K'' are realized inside the space of continuous functions on ''K''.
The SU(2) case
We now consider the case of the compact group SU(2). The representations are often considered from the
Lie algebra point of view, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form
:
According to the example discussed above in the section on representations of ''T'', the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers
. The general theory then tells us that for each
, there is a unique irreducible representation of SU(2) with highest weight
.
Much information about the representation corresponding to a given
is encoded in its character. Now, the Weyl character formula says,
in this case, that the character is given by
:
We can also write the character as sum of exponentials as follows:
:
(If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.)
From this last expression and the standard formula for the
character in terms of the weights of the representation, we can read off that the weights of the representation are
:
each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are
weights, each with multiplicity 1, the dimension of the representation is
. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.
An outline of the proof
We now outline the proof of the theorem of the highest weight, following the original argument of
Hermann Weyl. We continue to let
be a connected compact Lie group and
a fixed maximal torus in
. We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.
[ Sections 12.4 and 12.5]
The tools for the proof are the following:
*The
torus theorem.
*The
Weyl integral formula.
*The
Peter–Weyl theorem for class functions, which states that the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions on
.
With these tools in hand, we proceed with the proof. The first major step in the argument is to prove the
Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by . There is a closely related formula for the ch ...
. The formula states that if
is an irreducible representation with highest weight
, then the character
of
satisfies:
:
for all
in the Lie algebra of
. Here
is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of
in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the
norm of the character is 1. Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1.
Next, we let
denote the function on the right-hand side of the character formula. We show that ''even if
is not known to be the highest weight of a representation'',
is a well-defined, Weyl-invariant function on
, which therefore extends to a class function on
. Then using the Weyl integral formula, one can show that as
ranges over the set of dominant, analytically integral elements, the functions
form an orthonormal family of class functions. We emphasize that we do not currently know that every such
is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions
, and these functions are orthonormal.
Now comes the conclusion. The set of all
—with
ranging over the dominant, analytically integral elements—forms an orthonormal set in the space of square integrable class functions. But by the Weyl character formula, the characters of the irreducible representations form a subset of the
's. And by the Peter–Weyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions. If there were some
that is not the highest weight of a representation, then the corresponding
would not be the character of a representation. Thus, the characters would be a ''proper'' subset of the set of
's. But then we have an impossible situation: an orthonormal ''basis'' (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of
's). Thus, every
must actually be the highest weight of a representation.
Duality
The topic of recovering a compact group from its representation theory is the subject of the
Tannaka–Krein duality
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologic ...
, now often recast in terms of
Tannakian category
In mathematics, a Tannakian category is a particular kind of monoidal category ''C'', equipped with some extra structure relative to a given field ''K''. The role of such categories ''C'' is to approximate, in some sense, the category of linear re ...
theory.
From compact to non-compact groups
The influence of the compact group theory on non-compact groups was formulated by Weyl in his
unitarian trick
In mathematics, the unitarian trick is a device in the representation theory of Lie groups, introduced by for the special linear group and by Hermann Weyl for general semisimple groups. It applies to show that the representation theory of some g ...
. Inside a general
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals and homomorphisms, i ...
there is a
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups.
Maximal compact subgroups play an important role in the class ...
, and the representation theory of such groups, developed largely by
Harish-Chandra
Harish-Chandra Fellow of the Royal Society, FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
...
, uses intensively the
restriction of a representation to such a subgroup, and also the model of Weyl's character theory.
See also
*
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, ...
*
Maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.
A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefor ...
*
Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representati ...
*
Locally compact group
In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are loc ...
*
''p''-compact group
*
Protorus
In mathematics, a protorus is a compact connected topological abelian group. Equivalently, it is a projective limit of tori (products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian gr ...
*
Classifying finite-dimensional representations of Lie algebras
*
Weights in the representation theory of semisimple Lie algebras
References
Bibliography
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{{Authority control
Topological groups
Lie groups
Fourier analysis