Topological group

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The real numbers form a topological group under addition

In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are group and topological spaces at the same time, such that the continuity condition for the group operations connect 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.

Formal definition[]

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

⋅ : G × G → G, (x, y) ↦ xy

and inversion map:

⁻¹ : G → G, x ↦ x ⁻¹

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.

Checking continuity

The product map is continuous if and only if for any x, y ∈ G and any neighborhood W of xy in G, there exist neighborhoods U of x and V of y in G such that U ⋅ V ⊆ W, where U ⋅ V := {u ⋅ v : u ∈ U, v ∈ V}. The inversion map is continuous if and only if for any x ∈ G and any neighborhood V of x ⁻¹ in G, there exists a neighborhood U of x in G such that U ⁻¹ ⊆ V, where U ⁻¹ := { u⁻¹ : u ∈ U }.

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

G × G → G, (x, y) ↦ xy ⁻¹

is continuous. Explicitly, this means that for any x, y ∈ G 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.

Additive notation

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

+ : G × G → G , (x, y) ↦ x + y

− : G → G , x ↦ −x.

Hausdorffness

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.

Category

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 → H. 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.

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

The real numbers, ℝ with the usual topology form a topological group under addition. Euclidean n-space ℝⁿ 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 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 that 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.

The group ℤ_p is a pro-finite group; it is isomorphic to a subgroup of the product ${\displaystyle \prod _{n\geq 1}\mathbb {Z} /p^{n}}$ in such a way that its topology is induced by the product topology, where the finite groups ${\displaystyle \mathbb {Z} /p^{n}}$ 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.

Properties[]

Translation invariance

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. Consequently, if