Topological group

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The real numbers form a topological group under addition

In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics.

Formal definition

A topological group, G, is a topological space which is also a group such that the group operations of product:

${\displaystyle G\times G\to G:(x,y)\mapsto xy}$

and taking inverses:

${\displaystyle G\to G:x\mapsto x^{-1}}$

are continuous. Here G × G is viewed as a topological space with the product topology.

Although not part of this definition, many authors^[1] require that the topology on G be Hausdorff; it is equivalent to assume that the singleton containing the identity element 1 is a closed subset of G. The reasons, and some equivalent conditions, are discussed below. In any case, any topological group can be made Hausdorff by taking an appropriate canonical quotient.

In the language of category theory, topological groups can be defined concisely as group objects in the category of topological spaces, in the same way that ordinary groups are group objects in the category of sets. Note that the axioms are given in terms of the maps (binary product, unary inverse, and nullary identity), hence are categorical definitions.

Homomorphisms

A homomorphism of topological groups means a continuous group homomorphism G ${\displaystyle \to }$ H. An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.

Topological groups, together with their homomorphisms, form a category.

Examples

Every group can be trivially made into a topological group by considering it with the discrete topology; such groups are called discrete groups. In this sense, the theory of topological groups subsumes that of ordinary groups.

The real numbers, R with the usual topology form a topological group under addition. More generally, Euclidean n-space Rⁿ is a topological group under addition. Some other examples of abelian topological groups are the circle group S¹, or the torus (S¹)ⁿ for any natural number n.

The classical groups are important examples of non-abelian topological groups. For instance, the general linear group GL(n,R) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL(n,R) as a subspace of Euclidean space R^n×n. Another classical group is the orthogonal group O(n), the group of all linear maps from Rⁿ to itself that preserve the length of all vectors. The orthogonal group is compact as a topological space. Much of Euclidean geometry can be viewed as studying the structure of the orthogonal group, or the closely related group O(n) ⋉ Rⁿ of isometries of Rⁿ.

The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous. Lie groups are the best-understood topological groups; many questions about Lie groups can be converted to purely algebraic questions about Lie algebras and then solved.

An example of a topological group which is not a Lie group is the additive group Q of rational numbers, with the topology inherited from R. This is a countable space, and it does not have the discrete topology. An important example for number theory is the group Z_p of p-adic integers, for a prime number p, meaning the inverse limit of the finite groups Z/pⁿ as n goes to infinity. The group Z_p is well behaved in that it is compact (in fact, homeomorphic to the Cantor set), but it differs from (real) Lie groups in that it is totally disconnected. More generally, there is a theory of p-adic Lie groups, including compact groups such as GL(n,Z_p) as well as locally compact groups such as GL(n,Q_p), where Q_p is the locally compact field of p-adic numbers.

Some topological groups can be viewed as infinite dimensional Lie groups; this phrase is best understood informally, to include several different families of examples. For example, a topological vector space, such as a Banach space or Hilbert space, is an abelian topological group under addition. Some other infinite-dimensional groups that have been studied, with varying degrees of success, are loop groups, Kac–Moody groups, diffeomorphism groups, homeomorphism groups, and gauge groups.

In every Banach algebra with multiplicative identity, the set of invertible elements forms a topological group under multiplication. For example, the group of invertible bounded operators on a Hilbert space arises this way.

Properties

The inversion operation on a topological group G is a homeomorphism from G to itself. Likewise, if a is any element of G, then left or right multiplication by a yields a homeomorphism G → G.

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.^[2] If G is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

As a uniform space, every topological group is completely regular. It follows that if the identity element is closed in a topological group G, then G is T₂ (Hausdorff), even T_3½ (Tychonoff). If G is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group G/K, where K is the closure of the identity.^[3] This is equivalent to taking the Kolmogorov quotient of G.

The Birkhoff–Kakutani theorem states that the following three conditions on a topological group G are equivalent:^[4]

• The identity element 1 is closed in G, and there is a countable basis of neighborhoods for 1 in G.
• G is metrizable (as a topological space).
• There is a left-invariant metric on G that induces the given topology on G. (A metric on G is called left-invariant if for each point a in G, the map x ↦ ax is an isometry from G to itself.)

Every subgroup of a topological group is itself a topological group when given the subspace topology. If H is a subgroup of G, the set of left cosets G/H with the quotient topology is called a homogeneous space for G. The quotient map q : G → G/H is always open. For example, for a positive integer n, the sphere Sⁿ is a homogeneous space for the rotation group SO(n+1) in Rⁿ⁺¹, with Sⁿ = SO(n+1)/SO(n). A homogeneous space G/H is Hausdorff if and only if H is closed in G.^[5] Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.

Every open subgroup H is also closed in G, since the complement of H is the open set given by the union of open sets gH for g in G \ H.

If H is a normal subgroup of G, then the quotient group G/H becomes a topological group when given the quotient topology. It is Hausdorff if and only if H is closed in G. For example, the quotient group R/Z is isomorphic to the circle group S¹.

If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup of G, the closure of H is normal in G.

In any topological group, the identity component (i.e., the connected component containing the identity element) is a closed normal subgroup. If C is the identity component and a is any point of G, then the left coset aC is the component of G containing a. So the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. It follows that the quotient group G/C is totally disconnected.^[6]

The isomorphism theorems from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups. The theorems are valid if one places certain restrictions on the maps involved. For example, the first isomorphism theorem states that if f : G → H is a homomorphism, then the homomorphism from G/ker(f) to im(f) is an isomorphism if and only if the map f is open onto its image.^[7]

Hilbert's fifth problem

There are several strong results on the relation between topological groups and Lie groups. First, every continuous homomorphism of Lie groups G → H is smooth. It follows that a topological group has a unique structure of a Lie group if one exists. Also, Cartan's theorem says that every closed subgroup of a Lie group is a Lie subgroup, in particular a smooth submanifold.

Hilbert's fifth problem asked whether a topological group G which is a topological manifold must be a Lie group. In other words, does G have the structure of a smooth manifold, making the group operations smooth? The answer is yes, by Gleason, Montgomery, and Zippin.^[8] In fact, G has a real analytic structure. Using the smooth structure, one can define the Lie algebra of G, an object of linear algebra which determines a connected group G up to covering spaces. As a result, the solution to Hilbert's fifth problem reduces the classification of topological groups that are topological manifolds to an algebraic problem, albeit a complicated problem in general.

The theorem also has consequences for broader classes of topological groups. First, every compact group (understood to be Hausdorff) is an inverse limit of compact Lie groups. (One important case is an inverse limit of finite groups, called a profinite group. For example, the group Z_p of p-adic integers and the absolute Galois group of a field are profinite groups.) Furthermore, every connected locally compact group is an inverse limit of connected Lie groups.^[9] At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group.^[10] (For example, the locally compact group GL(n,Q_p) contains the compact open subgroup GL(n,Z_p), which is the inverse limit of the finite groups GL(n,Z/p^r) as r goes to infinity.)

Representations of compact or locally compact groups

An action of a topological group G on a topological space X is a group action of G on X such that the corresponding function G × X → X is continuous. Likewise, a representation of a topological group G on a real or complex topological vector space V is a continuous action of G on V such that for each g in G, the map v ↦ gv from V to itself is linear.

Group actions and representation theory are particularly well understood for compact groups, generalizing what happens for finite groups. For example, every finite-dimensional (real or complex) representation of a compact group is a direct sum of irreducible representations. An infinite-dimensional unitary representation of a compact group can be decomposed as a Hilbert-space direct sum of irreducible representations, which are all finite-dimensional; this is part of the Peter–Weyl theorem.^[11] For example, the theory of Fourier series describes the decomposition of the unitary representation of the circle group S¹ on the complex Hilbert space L²(S¹). The irreducible representations of S¹ are all 1-dimensional, of the form z ↦ zⁿ for integers n (where S¹ is viewed as a subgroup of the multiplicative group C*). Each of these representations occurs with multiplicity 1 in L²(S¹).

The irreducible representations of all compact connected Lie groups have been classified. In particular, the character of each irreducible representation is given by the Weyl character formula.

More generally, locally compact groups have a rich theory of harmonic analysis, because they admit a natural notion of measure and integral, given by the Haar measure. Every unitary representation of a locally compact group can be described as a direct integral of irreducible unitary representations. (The decomposition is essentially unique if G is of Type I, which includes the most important examples such as abelian groups and semisimple Lie groups.^[12]) A basic example is the Fourier transform, which decomposes the action of the additive group R on the Hilbert space L²(R) as a direct integral of the irreducible unitary representations of R. The irreducible unitary representations of R are all 1-dimensional, of the form x ↦ e^2πiax for a ∈ R.

The irreducible unitary representations of a locally compact group may be infinite-dimensional. A major goal of representation theory, related to the Langlands classification of admissible representations, is to find the unitary dual (the space of all irreducible unitary representations) for the semisimple Lie groups. The unitary dual is known in many cases such as SL(2,R), but not all.

For a locally compact abelian group G, every irreducible unitary representation has dimension 1. In this case, the unitary dual ${\displaystyle {\hat {G}}}$ is a group, in fact another locally compact abelian group. Pontryagin duality states that for a locally compact abelian group G, the dual of ${\displaystyle {\hat {G}}}$ is the original group G. For example, the dual group of the integers Z is the circle group S¹, while the group R of real numbers is isomorphic to its own dual.

Every locally compact group G has a good supply of irreducible unitary representations; for example, enough representations to distinguish the points of G (the Gelfand–Raikov theorem). By contrast, representation theory for topological groups that are not locally compact has so far been developed only in special situations, and it may not be reasonable to expect a general theory. For example, there are many abelian Banach–Lie groups for which every representation on Hilbert space is trivial.^[13]

Homotopy theory of topological groups

Topological groups are special among all topological spaces, even in terms of their homotopy type. One basic point is that a topological group G determines a path-connected topological space, the classifying space BG (which classifies principal G-bundles over topological spaces, under mild hypotheses). The group G is isomorphic in the homotopy category to the loop space of BG; that implies various restrictions on the homotopy type of G.^[14] Some of these restrictions hold in the broader context of H-spaces.

For example, the fundamental group of a topological group G is abelian. (More generally, the Whitehead product on the homotopy groups of G is zero.) Also, for any field k, the cohomology ring H*(G,k) has the structure of a Hopf algebra. In view of structure theorems on Hopf algebras by Heinz Hopf and Armand Borel, this puts strong restrictions on the possible cohomology rings of topological groups. In particular, if G is a path-connected topological group whose rational cohomology ring H*(G,Q) is finite-dimensional in each degree, then this ring must be a free graded-commutative algebra over Q, that is, the tensor product of a polynomial ring on generators of even degree with an exterior algebra on generators of odd degree.^[15]

In particular, for a connected Lie group G, the rational cohomology ring of G is an exterior algebra on generators of odd degree. Moreover, a connected Lie group G has a maximal compact subgroup K, which is unique up to conjugation, and the inclusion of K into G is a homotopy equivalence. So describing the homotopy types of Lie groups reduces to the case of compact Lie groups. For example, the maximal compact subgroup of SL(2,R) is the circle group SO(2), and the homogeneous space SL(2,R)/SO(2) can be identified with the hyperbolic plane. Since the hyperbolic plane is contractible, the inclusion of the circle group into SL(2,R) is a homotopy equivalence.

Finally, compact connected Lie groups have been classified by Wilhelm Killing, Élie Cartan, and Hermann Weyl. As a result, there is an essentially complete description of the possible homotopy types of Lie groups. For example, a compact connected Lie group of dimension at most 3 is either a torus, the group SU(2) (diffeomorphic to the 3-sphere S³), or its quotient group SU(2)/{±1} ≅ SO(3) (diffeomorphic to RP³).

Generalizations

Various generalizations of topological groups can be obtained by weakening the continuity conditions:^[16]

• A semitopological group is a group G with a topology such that for each c in G the two functions G → G defined by ${\displaystyle x\mapsto xc}$ and ${\displaystyle x\mapsto cx}$ are continuous.
• A quasitopological group is a semitopological group in which the function mapping elements to their inverses is also continuous.
• A paratopological group is a group with a topology such that the group operation is continuous.