# circle group

In mathematics, the circle group, denoted by T (or in blackboard bold by ), is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane.
The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers. Since C× is abelian, it follows that T is as well.

The notation T for the circle group stems from the fact that Tn (the direct product of T with itself n times) is geometrically an n-torus. The circle group is then a 1-torus.

The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.

## Elementary introduction

One way to think about the circle group is that it describes how to add angles, where only angles between 0Â° and 360Â° are permitted. For example, the diagram illustrates how to add 150Â° to 270Â°. The answer should be 150Â° + 270Â° = 420Â°, but when thinking in terms of the circle group, we need to "forget" the fact that we have wrapped once around the circle. Therefore we adjust our answer by 360Â° which gives 420Â° − 360Â° = 60Â°.

Another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed. To achieve this, we might need to throw away digits occurring before the decimal point. For example, when we work out 0.784 + 0.925 + 0.446, the answer should be 2.155, but we throw away the leading 2, so the answer (in the circle group) is just 0.155.

## Topological and analytic structure

The circle group is more than just an abstract algebraic group. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on C×, the circle group has the structure of a topological group. Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of C× (itself regarded as a topological group).

One can say even more. The circle is a 1-dimensional real manifold and multiplication and inversion are real-analytic maps on the circle. This gives the circle group the structure of a 1-dimensional Lie group. In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group. Moreover, every n-dimensional compact, connected, abelian Lie group is isomorphic to Tn.

## Isomorphisms

The circle group shows up in a huge variety of forms in mathematics. We list some of the more common forms here. Specifically, we show that

The set of all 1×1 unitary matrices clearly coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1. Therefore, the circle group is canonically isomorphic to U(1), the first unitary group.

The exponential function gives rise to a group homomorphism exp : RT from the additive real numbers R to the circle group T via the map

The last equality is Euler's formula. The real number θ corresponds to the angle on the unit circle as measured from the positive x-axis. That this map is a homomorphism follows from the fact the multiplication of unit complex numbers corresponds to addition of angles:

This exponential map is clearly a surjective function from R to T. It is not, however, injective. The kernel of this map is the set of all integer multiples of 2π. By the first isomorphism theorem we then have that

After rescaling we can also say that T is isomorphic to R/Z.

If complex numbers are realized as 2×2 real matrices (see complex number), the unit complex numbers correspond to 2×2 orthogonal matrices with unit determinant. Specifically, we have

The circle group is therefore isomorphic to the special orthogonal group SO(2). This has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex plane, and every such rotation is of this form.

## Properties

Any compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group. That means that, thinking in terms of symmetry, a compact symmetry group acting continuously can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen for example at rotational invariance, and spontaneous symmetry breaking.

The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer n > 0, the nth roots of unity form a cyclic group of order n, which is unique up to isomorphism.

## Representations

The representations of the circle group are easy to describe. It follows from Schur's lemma that the irreducible complex representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation ρ : TGL(1, C) ≅ C×, must take values in U(1)≅ T. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself. Every such homomorphism is of the form

These representations are all inequivalent. The representation φ-n is conjugate to φn,

These representations are just the characters of the circle group. The character group of T is clearly an infinite cyclic group generated by φ1:

The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representations
taking values in SO(2). Here we only have positive integers n since the representation is equivalent to .

## Algebraic structure

In this section we will forget about the topological structure of the circle group and look only at its algebraic structure.

The circle group T is a divisible group. Its torsion subgroup is given by the set of all nth roots of unity for all n, and is isomorphic to Q/Z. The structure theorem for divisible groups tells us that T is isomorphic to the direct sum of Q/Z with a number of copies of Q. The number of copies of Q must be c (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of c copies of Q is isomorphic to R, as R is a vector space of dimension c over Q. Thus

The isomorphism

can be proved in the same way, as C× is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of T.

## References

Circle was an early 1970s avant garde jazz ensemble. Its core members were
• Chick Corea, piano
• Dave Holland, bass
• Barry Altschul, drums and percussion.
For some of its existence it also included Anthony Braxton in a leading role on several reed instruments.
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
Blackboard bold is a typeface style often used for certain symbols in mathematics and physics texts, in which certain lines of the symbol (usually vertical, or near-vertical lines) are doubled. The symbols usually describe number sets.
group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
In mathematics, a complex number is a number of the form

where a and b are real numbers, and i is the imaginary unit, with the property i Â² = −1.
unit circle is a circle with a unit radius, i.e., a circle whose radius is 1. Frequently, especially in trigonometry, "the" unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Euclidean plane.
subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group operation on H.
In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that the group operation * is commutative, so that for all a and b in G, a * b = b * a.
In mathematics, one can often define a direct product of objects already known, giving a new one. Examples are the product of sets (see Cartesian product), groups (described below), the product of rings and of other algebraic structures.
In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle. Examples of tori include the surfaces of doughnuts and inner tubes.
Pontryagin duality explains the general properties of the Fourier transform. It places in a unified context a number of observations about functions on the real line or on finite abelian groups:

In mathematics, a Lie group (IPA pronunciation: [liː], sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. Intuitively, this is a function f where a set of points near f(x) always contain the image of a set of points near x.
In mathematics, a topological group is a group G together with a topology on G such that the group and topological structures are compatible. Specifically, this means that the group operations are continuous functions.
In topology and related branches of mathematics, a closed set is a set whose complement is open.

## Definition of a closed set

In a metric space, a set is closed if every limit point of the set is a point in the set.
manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important.
In mathematics, a Lie group (IPA pronunciation: [liː], sounds like "Lee"), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
In mathematics, the phrase "up to xxxx" indicates that members of an equivalence class are to be regarded as a single entity for some purpose. "xxxx" describes a property or process which transforms an element into one from the same equivalence class, i.e.
Rn is called compact if it is closed and bounded. For example, in R, the closed unit interval [0, 1] is compact, but the set of integers Z is not (it is not bounded) and neither is the half-open interval [0, 1) (it is not closed).
In topology and related branches of mathematics, a connected space is a topological space which cannot be represented as the disjoint union of two or more nonempty open spaces.
In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition

where is the identity matrix and is the conjugate transpose (also called the Hermitian adjoint) of U.
In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication.
The exponential function is one of the most important functions in mathematics. The application of this function to a value x is written as exp(x).
In mathematics, given two groups (G, *) and (H, Â·), a group homomorphism from (G, *) to (H, Â·) is a function h : GH such that for all u and v in G it holds that

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. (Euler's identity is a special case of the Euler formula.