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 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.

A topological group, G, is a topological space that is also a group such that the group operation (in this case product):

are continuous.[note 1] Here G × G is viewed as a topological space with the product topology. Such a topology is said to be compatible with the group operations and is called a group topology.

To show that a topology is compatible with the group operations, it suffices to check that the map

is continuous. Explicitly, this means that for any x, yG and any neighborhood W in G of xy−1, there exist neighborhoods U of x and V of y in G such that U ⋅ (V−1) ⊆ W.

This definition used notation for multiplicative groups; the equivalent for additive groups would be that the following two operations are continuous:

Although not part of this definition, many authors[3] require that the topology on G be Hausdorff. One reason for this is that any topological group can be canonically associated with a Hausdorff topological group by taking an appropriate canonical quotient; this however, often still requires working with the original non-Hausdorff topological group. Other reasons, and some equivalent conditions, are discussed below.

This article will not assume that topological groups are necessarily Hausdorff.

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.

A homomorphism of topological groups means a continuous group homomorphism GH. Topological groups, together with their homomorphisms, form a category. A group homomorphism between commutative topological groups is continuous if and only if it is continuous at some point.[4]

An isomorphism of topological groups is a group isomorphism that 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 that 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.

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 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.

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.

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 MN, 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 MN (where cl M is symmetric as well).[4]

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.

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 KNU.[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]

A subgroup of a commutative topological group is discrete if and only if it has an isolated point.[4]

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.[6] This is equivalent to taking the Kolmogorov quotient of G.

The Birkhoff–Kakutani theorem (named after mathematicians Garrett Birkhoff and Shizuo Kakutani) states that the following three conditions on a topological group G are equivalent:[7]

Furthermore, the following are equivalent for any topological group G:

Every subgroup of a topological group is itself a topological group when given the subspace topology. 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 gG \ H. 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.[11]

In any commutative topological group, the product (assuming the group is multiplicative) KC of a compact set K and a closed set C is a closed set.[4] Furthermore, for any subsets R and S of G, (cl R)(cl S) ⊆ cl (RS).[4]

If H is a subgroup of a commutative topological group G and if N is a neighborhood in G of the identity element such that H ∩ cl N is closed, then H is closed.[4] Every discrete subgroup of a Hausdorff commutative topological group is closed.[4]

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 third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.

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.

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 × XX 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 gG, the map vgv from V to itself is linear.

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.

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 , but not all.

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]

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.

Information about convergence of nets and filters, such as definitions and properties, can be found in the article about filters in topology.

A topological group is called sequentially complete if it is a sequentially complete subset of itself.

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