Topological group

In mathematics, topological groups are 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 structures together and consequently they are not independent from each other.^[1]

Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups.^[2]

Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis.

Examples[edit]

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 indiscrete topology (i.e. the trivial topology) also makes every group into a topological group.

The real numbers, $\mathbb {R}$ with the usual topology form a topological group under addition. Euclidean $n$ -space $\mathbb {R}$ is also a topological group under addition, and more generally, every topological vector space forms an (abelian) topological group. Some other examples of abelian topological groups are the circle group $S 1$ , or the torus $(S 1) n$ for any natural number $n$ .

The classical groups are important examples of non-abelian topological groups. For instance, the general linear group $\mathbb {R}$ of all invertible $n$ -by- $n$ matrices with real entries can be viewed as a topological group with the topology defined by viewing $\mathbb {R}$ as a subspace of Euclidean space $\mathbb {R}$ . Another classical group is the orthogonal group $O(n)$ , the group of all linear maps from $\mathbb {R}$ 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 $\mathbb {R}$ of isometries of $\mathbb {R}$ .

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 that is not a Lie group is the additive group $\mathbb {Q}$ of rational numbers, with the topology inherited from $\mathbb {R}$ . This is a countable space, and it does not have the discrete topology. An important example for number theory is the group $\mathbb {Z}$ of p-adic integers, for a prime number $p$ , meaning the inverse limit of the finite groups $\mathbb {Z}$ as n goes to infinity. The group $\mathbb {Z}$ 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 $\mathbb {Z}$ as well as locally compact groups such as $\mathbb {Q}$ , where $\mathbb {Q}$ is the locally compact field of p-adic numbers.

The group $\mathbb {Z}$ is a pro-finite group; it is isomorphic to a subgroup of the product $\prod _{n\geq 1}\mathbb {Z} /p^{n}$ in such a way that its topology is induced by the product topology, where the finite groups $\mathbb {Z} /p^{n}$ are given the discrete topology. Another large class of pro-finite groups important in number theory are absolute Galois groups.

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.

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Properties[edit]

Translation invariance[edit]

Every topological group's topology is translation invariant, which by definition means that if for any $a\in G,$ left or right multiplication by this element yields a homeomorphism $G\to G.$ Consequently, for any $a\in G$ and $S\subseteq G,$ the subset $S$ is open (resp. closed) in $G$ if and only if this is true of its left translation $aS:=\{as:s\in S\}$ and right translation $Sa:=\{sa:s\in S\}.$ If ${\mathcal {N}}$ is a neighborhood basis of the identity element in a topological group $G$ then for all $x\in X,$ $x{\mathcal {N}}:=\{xN:N\in {\mathcal {N}}\}$ is a neighborhood basis of $x$ in $G.$ ^[4] In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element. If $S$ is any subset of $G$ and $U$ is an open subset of $G,$ then $SU:=\{su:s\in S,u\in U\}$ is an open subset of $G.$ ^[4]

Symmetric neighborhoods[edit]

The inversion operation $g\mapsto g^{-1}$ on a topological group $G$ is a homeomorphism from $G$ to itself.

A subset $S\subseteq G$ is said to be symmetric if $S^{-1}=S,$ where $S^{-1}:=\left\{s^{-1}:s\in S\right\}.$ The closure of every symmetric set in a commutative topological group is symmetric.^[4] If $S$ is any subset of a commutative topological group $G$ , then the following sets are also symmetric: $S -1 \cap S$ , $S -1 \cup S$ , and $S -1 S$ .^[4]

For any neighborhood $N$ in a commutative topological group $G$ of the identity element, there exists a symmetric neighborhood $M$ of the identity element such that $M -1 M \subseteq N$ , where note that $M -1 M$ is necessarily a symmetric neighborhood of the identity element.^[4] Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.

If $G$ is a locally compact commutative group, then for any neighborhood $N$ in $G$ of the identity element, there exists a symmetric relatively compact neighborhood $M$ of the identity element such that $cl M \subseteq N$ (where $cl M$ is symmetric as well).^[4]

Uniform space[edit]

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.^[5] 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.

Separation properties[edit]

If $U$ is an open subset of a commutative topological group $G$ and $U$ contains a compact set $K$ , then there exists a neighborhood $N$ of the identity element such that $KN \subseteq U$ .^[4]

As a uniform space, every commutative topological group is completely regular. Consequently, for a multiplicative topological group $G$ with identity element 1, the following are equivalent:^[4]

Representations of compact or locally compact groups[edit]

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 $G \times X \to X$ 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 $v \mapsto gv$ 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.^[16] For example, the theory of Fourier series describes the decomposition of the unitary representation of the circle group $S 1$ on the complex Hilbert space $L 2 (S 1)$ . The irreducible representations of $S 1$ are all 1-dimensional, of the form $z \mapsto z n$ for integers $n$ (where $S 1$ is viewed as a subgroup of the multiplicative group $\mathbb {C}$ *). Each of these representations occurs with multiplicity 1 in $L 2 (S 1)$ .

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.^[17]) A basic example is the Fourier transform, which decomposes the action of the additive group $\mathbb {R}$ on the Hilbert space $\mathbb {R}$ as a direct integral of the irreducible unitary representations of $\mathbb {R}$ . The irreducible unitary representations of $\mathbb {R}$ are all 1-dimensional, of the form $x \mapsto e 2π iax$ for $\mathbb {R}$ .

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 $\mathbb {R}$ , but not all.

For a locally compact abelian group $G$ , every irreducible unitary representation has dimension 1. In this case, the unitary dual ${\hat {G}}$ is a group, in fact another locally compact abelian group. Pontryagin duality states that for a locally compact abelian group $G$ , the dual of ${\hat {G}}$ is the original group $G$ . For example, the dual group of the integers $\mathbb {Z}$ is the circle group $S 1$ , while the group $\mathbb {R}$ 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.^[18]

Hilbert's fifth problem[edit]

There are several strong results on the relation between topological groups and Lie groups. First, every continuous homomorphism of Lie groups $G\to H$ 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 $G$ that 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? As shown by Andrew Gleason, Deane Montgomery, and Leo Zippin, the answer to this problem is yes.^[13] 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 that 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 $\mathbb {Z}$ 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.^[14] At the other extreme, a totally disconnected locally compact group always contains a compact open subgroup, which is necessarily a profinite group.^[15] (For example, the locally compact group $\mathbb {Q}$ contains the compact open subgroup $\mathbb {Z}$ , which is the inverse limit of the finite groups $\mathbb {Z}$ as $r$ ' goes to infinity.)

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Homotopy theory of topological groups[edit]

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 $G$ .^[19] 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 $\mathbb {Q}$ is finite-dimensional in each degree, then this ring must be a free graded-commutative algebra over $\mathbb {Q}$ , that is, the tensor product of a polynomial ring on generators of even degree with an exterior algebra on generators of odd degree.^[20]

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 $\mathbb {R}$ is the circle group $SO(2)$ , and the homogeneous space $\mathbb {R}$ can be identified with the hyperbolic plane. Since the hyperbolic plane is contractible, the inclusion of the circle group into $\mathbb {R}$ 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 $S 3$ ), or its quotient group $SU(2)/{\pm1} ≅ SO(3)$ (diffeomorphic to $RP 3$ ).

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The canonical uniformity on any commutative topological group is translation-invariant.

The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin.

Every entourage $\Delta _{X}(N)$ contains the diagonal $\Delta _{X}:=\Delta _{X}(\{0\})=\{(x,x):x\in X\}$ because $0\in N.$

If $N$ is (that is, $-N=N$ ) then $\Delta _{X}(N)$ is symmetric (meaning that $\Delta _{X}(N)^{\operatorname {op} }=\Delta _{X}(N)$ ) and $\Delta _{X}(N)\circ \Delta _{X}(N)=\{(x,z):{\text{ there exists }}y\in X{\text{ such that }}x,z\in y+N\}=\bigcup _{y\in X}[(y+N)\times (y+N)]=\Delta _{X}+(N\times N).$

symmetric

The topology induced on $X$ by the canonical uniformity is the same as the topology that $X$ started with (that is, it is $\tau$ ).

A is a group $G$ with a topology such that for each $c \in G$ the two functions $G \to G$ defined by $x \mapsto xc$ and $x \mapsto cx$ are continuous.

semitopological group

A is a semitopological group in which the function mapping elements to their inverses is also continuous.

quasitopological group

A is a group with a topology such that the group operation is continuous.

paratopological group

Various generalizations of topological groups can be obtained by weakening the continuity conditions:^[26]

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– Algebraic variety with a group structure

Algebraic group

– algebraic structure that is complete relative to a metric

Complete field

– Topological group with compact topology

Compact group

– A TVS where points that get progressively closer to each other will always converge to a point

Complete topological vector space

– Group that is also a differentiable manifold with group operations that are smooth

Lie group

Locally compact field

– topological group G for which the underlying topology is locally compact and Hausdorff, so that the Haar measure can be defined

Locally compact group

– relatively new C*-algebraic approach toward quantum groups

Locally compact quantum group

– Topological group that is in a certain sense assembled from a system of finite groups

Topological group

Examples[edit]

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Properties[edit]

Translation invariance[edit]

Symmetric neighborhoods[edit]

Uniform space[edit]

Separation properties[edit]

Representations of compact or locally compact groups[edit]

Hilbert's fifth problem[edit]

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Homotopy theory of topological groups[edit]

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symmetric

semitopological group

quasitopological group

paratopological group

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Algebraic group

Complete field

Compact group

Complete topological vector space

Lie group

Locally compact field

Locally compact group

Locally compact quantum group

Profinite group

Ordered topological vector space

Topological abelian group

Topological field

Topological module

Topological ring

Topological semigroup

Topological vector space