View source for Circle group

{{Short description|Lie group of complex numbers of unit modulus; topologically a circle}}
{{other uses|Circle (disambiguation)}}
[[Image:Circle-group.svg|thumb|Multiplication on the circle group is equivalent to addition of angles.]]
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In [[mathematics]], the '''circle group''', denoted by <math>\mathbb T</math> or {{tmath|1= \mathbb S^1 }}, is the [[multiplicative group]] of all [[complex number]]s with [[Absolute value#Complex numbers|absolute value]] 1, that is, the [[unit circle]] in the [[complex plane]] or simply the '''unit complex numbers'''<ref>{{cite book |last1=James |first1=Robert C. |author-link=Robert C. James |last2=James |first2=Glenn |year=1992 |title=Mathematics Dictionary |edition=Fifth |publisher=Chapman & Hall |page=436 |isbn=9780412990410 |url=https://books.google.com/books?id=UyIfgBIwLMQC&q=%22unit+complex+number%22&pg=PA436 |quote=a ''unit complex number'' is a [[complex number]] of [[1|unit]] [[absolute value]]}}.</ref>
<math display=block>\mathbb T = \{ z \in \C : |z| = 1 \}.</math>

The circle group forms a [[subgroup]] of {{tmath|1= \C^\times }}, the multiplicative group of all nonzero complex numbers. Since <math>\C^\times</math> is [[abelian group|abelian]], it follows that <math>\mathbb T</math> is as well.

A unit complex number in the circle group represents a [[rotation (mathematics)|rotation]] of the complex plane about the origin and can be parametrized by the [[angle measure]] {{tmath|1= \theta }}:
<math display=block>\theta \mapsto z = e^{i\theta} = \cos\theta + i\sin\theta.</math>

This is the [[exponential map (Lie theory)|exponential map]] for the circle group.

The circle group plays a central role in [[Pontryagin duality]] and in the theory of [[Lie group]]s.

The notation <math>\mathbb T</math> for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-[[torus]]. More generally, <math>\mathbb T^n</math> (the [[direct product of groups|direct product]] of <math>\mathbb T</math> with itself <math>n</math> times) is geometrically an <math>n</math>-torus.

The circle group is [[Group_isomorphism|isomorphic]] to the [[special orthogonal group]] {{tmath|1= \mathrm{SO}(2) }}.

== Elementary introduction ==
<!-- this section is intended to be accessible to a curious high school student -->

One way to think about the circle group is that it describes how to add ''angles'', where only angles between 0° and 360° or <math>\in[0, 2\pi)</math> or <math>\in(-\pi,+\pi]</math> are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer is {{nowrap|150° + 270° {{=}} 420°}}, but when thinking in terms of the circle group, we may "forget" the fact that we have wrapped once around the circle. Therefore, we adjust our answer by 360°, which gives {{nowrap|420° ≡ 60° ([[modular arithmetic|mod]] 360°}}).

Another description is in terms of ordinary (real) addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation: 360° or {{tmath|1= 2\pi }}), i.e. the real numbers modulo the integers: {{tmath|1= \mathbb T \cong \R/\Z }}. This can be achieved by throwing away the digits occurring before the decimal point. For example, when we work out {{nowrap|0.4166... + 0.75}}, the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just {{tmath|1= 0.1\bar{6} \equiv 1.1\bar{6} \equiv -0.8\bar{3}\;(\text{mod}\,\Z) }}, with some preference to 0.166..., because {{tmath|1= 0.1\bar{6} \in [0,1) }}.

== Topological and analytic structure ==
The circle group is more than just an abstract algebraic object. It has a [[natural topology]] when regarded as a [[subspace (topology)|subspace]] of the complex plane. Since multiplication and inversion are [[continuous function (topology)|continuous functions]] on {{tmath|1= \C^\times }}, 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 <math>\C^\times</math> (itself regarded as a topological group).

One can say even more. The circle is a 1-dimensional real [[manifold]], and multiplication and inversion are [[analytic function|real-analytic maps]] on the circle. This gives the circle group the structure of a [[one-parameter group]], an instance of a [[Lie group]]. In fact, [[up to]] isomorphism, it is the unique 1-dimensional [[compact space|compact]], [[connected space|connected]] Lie group. Moreover, every <math>n</math>-dimensional compact, connected, abelian Lie group is isomorphic to {{tmath|1= \mathbb T^n }}.

== Isomorphisms ==
The circle group shows up in a variety of forms in mathematics. We list some of the more common forms here. Specifically, we show that
<math display=block>\mathbb T \cong \mbox{U}(1) \cong \R/\Z \cong \mathrm{SO}(2),</math>
where the slash ({{tmath|~\!/~\!}}) denotes [[quotient group|group quotient]] and <math>\cong</math> the existence of an [[isomorphism]] between the groups.

The set of all {{tmath|1\times1}} [[unitary matrix|unitary matrices]] coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value&nbsp;1. Therefore, the circle group is canonically isomorphic to the first [[unitary group]] {{tmath|1= \mathrm{U}(1) }}, i.e.,
<math display=block>\mathbb T \cong \mbox{U}(1).</math>
The [[exponential function]] gives rise to a map <math>\exp : \R \to \mathbb T</math> from the additive real numbers {{tmath|\R}} to the circle group {{tmath|\mathbb T}} known as [[Euler's formula]]
<math display=block>\theta \mapsto e^{i\theta} = \cos\theta + i \sin \theta,</math>
where <math>\theta \in \mathbb{R}</math> corresponds to the angle (in [[radian]]s) on the unit circle as measured counterclockwise from the positive ''x''-axis. The property
<math display=block>e^{i\theta_1} e^{i\theta_2} = e^{i(\theta_1+\theta_2)}, \quad \forall \theta_1 ,\theta_2 \in \mathbb{R},</math>
makes <math>\exp : \R \to \mathbb T</math> a [[group homomorphism]]. While the map is [[surjective]], it is not [[injective]] and therefore not an isomorphism. The [[kernel (group theory)|kernel]] of this map is the set of all [[integer]] multiples of {{tmath|1= 2\pi }}. By the [[first isomorphism theorem]] we then have that
<math display=block>\mathbb T \cong \R~\!/~\!2\pi\Z.</math>
After rescaling we can also say that <math>\mathbb T</math> is isomorphic to {{tmath|1= \R/\Z }}.

The unit [[Complex_number#Matrix_representation_of_complex_numbers|complex numbers]] can be realized as 2×2 real [[orthogonal matrices]], i.e.,
<math display=block> e^{i\theta}= \cos\theta + i \sin \theta \leftrightarrow \begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta &  \cos \theta \\
\end{bmatrix} = f\bigl(e^{i\theta}\bigr),
</math>
associating the [[Absolute_value#Complex_numbers|squared modulus]] and [[complex conjugate]] with the [[determinant]] and [[transpose]], respectively, of the corresponding matrix. As the [[List_of_trigonometric_identities#Angle_sum_and_difference_identities|angle sum trigonometric identities]] imply that
<math display=block>
  f\bigl(e^{i\theta_1} e^{i\theta_2}\bigr) = \begin{bmatrix}
    \cos(\theta_1 + \theta_2) & -\sin(\theta_1 + \theta_2) \\
    \sin(\theta_1 + \theta_2) &  \cos(\theta_1 + \theta_2)
  \end{bmatrix} =
  f\bigl(e^{i\theta_1}\bigr) \times f\bigl(e^{i\theta_2}\bigr),
</math>
where <math>\times</math> is matrix multiplication, the circle group is [[Group_isomorphism|isomorphic]] to the [[special orthogonal group]] <math>\mathrm{SO}(2)</math>, i.e.,
<math display=block>\mathbb T \cong \mathrm{SO}(2).</math>
This isomorphism has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex (and real) plane, and every such rotation is of this form.

== Properties ==
Every compact Lie group <math>\mathrm{G}</math> of dimension&nbsp;>&nbsp;0 has a [[subgroup]] isomorphic to the circle group. This 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 [[subgroup]]s, but its only proper [[closed set|closed]] subgroups consist of [[root of unity|roots of unity]]: For each integer {{tmath|1= n > 0 }}, the <math>n</math>th roots of unity form a [[cyclic group]] of order&nbsp;{{tmath|1= n }}, which is unique up to isomorphism.

In the same way that the [[real numbers]] are a [[Completeness (topology)|completion]] of the [[dyadic rationals|''b''-adic rationals]] <math>\Z\bigl[\tfrac1b\bigr]</math> for every [[natural number]] {{tmath|1= b > 1 }}, the circle group is the completion of the [[Prüfer group]] <math>\Z\bigl[\tfrac{1}{b}\bigr]~\!/~\!\Z</math> for {{tmath|1= b }}, given by the [[direct limit]] {{tmath|1= \varinjlim \Z~\!/~\!b^n \Z }}.

== Representations ==
The [[group representation|representations]] of the circle group are easy to describe. It follows from [[Schur's lemma]] that the [[irreducible representation|irreducible]] [[complex number|complex]] representations of an abelian group are all 1-dimensional. Since the circle group is compact, any representation
<math display=block>\rho: \mathbb T \to \mathrm{GL}(1, \C) \cong \C^\times</math>
must take values in {{tmath|1= \mbox{U}(1) \cong \mathbb T }}. Therefore, the irreducible representations of the circle group are just the [[group homomorphism|homomorphisms]] from the circle group to itself.

For each integer <math>n</math> we can define a representation <math>\phi_n</math> of the circle group by {{tmath|1= \phi_n(z) = z^n }}.  These representations are all inequivalent. The representation <math>\phi_{-n}</math> is [[conjugate representation|conjugate]] to {{tmath|1= \phi_{n} }}:
<math display=block>\phi_{-n} = \overline{\phi_n}.</math>

These representations are just the [[character (mathematics)|characters]] of the circle group. The [[character group]] of <math>\mathbb T</math> is clearly an [[infinite cyclic group]] generated by {{tmath|1= \phi_1 }}:
<math display=block>\operatorname{Hom}(\mathbb T, \mathbb T) \cong \Z.</math>

The irreducible [[real number|real]] representations of the circle group are the [[trivial representation]] (which is 1-dimensional) and the representations
<math display=block>\rho_n\bigl(e^{i\theta}\bigr) = \begin{bmatrix}
 \cos n\theta & -\sin n\theta \\
 \sin n\theta & \cos n\theta
\end{bmatrix}, \quad
n \in \Z^+ ,</math>
taking values in {{tmath|1= \mathrm{SO}(2) }}. Here we only have positive integers {{tmath|1= n }}, since the representation <math>\rho_{-n}</math> is equivalent to {{tmath|1= \rho_n }}.

== Group structure ==
The circle group <math>\mathbb T</math> is a [[divisible group]]. Its [[torsion subgroup]] is given by the set of all <math>n</math>-th [[roots of unity]] for all <math>n</math> and is isomorphic to {{tmath|1= \Q/\Z }}. The [[Divisible group#Structure theorem of divisible groups|structure theorem]] for divisible groups and the [[axiom of choice]] together tell us that <math>\mathbb T</math> is isomorphic to the [[direct sum of abelian groups|direct sum]] of <math>\Q/\Z</math> with a number of copies of {{tmath|\Q}}.<ref>
{{cite book
 | last = Fuchs | first = László
 | contribution = Example 3.5
 | doi = 10.1007/978-3-319-19422-6
 | isbn = 978-3-319-19421-9
 | mr = 3467030
 | page = 141
 | publisher = Springer, Cham
 | series = Springer Monographs in Mathematics
 | title = Abelian groups
 | year = 2015
}}</ref>

The number of copies of {{tmath|\Q}} must be <math>\mathfrak c</math> (the [[cardinality of the continuum]]) in order for the cardinality of the direct sum to be correct. But the direct sum of <math>\mathfrak c</math> copies of {{tmath|\Q}} is isomorphic to {{tmath|\R}}, as <math>\R</math> is a [[vector space]] of dimension <math>\mathfrak c</math> over {{tmath|\Q}}. Thus,
<math display=block>\mathbb T \cong \R \oplus (\Q/\Z).</math>

The isomorphism
<math display=block>\C^\times \cong \R \oplus (\Q/\Z)</math>
can be proved in the same way, since {{tmath|\C^\times}} is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of {{tmath|\mathbb T}}.

== See also ==
{{Portal|Mathematics}}
* [[Group of rational points on the unit circle]]
* [[One-parameter subgroup]]
* [[n-sphere|{{mvar|n}}-sphere]]
* [[Orthogonal group]]
* [[Phase factor]] (application in quantum-mechanics)
* [[Rotation number]]
* [[Solenoid (mathematics)|Solenoid]]

== Notes ==
{{reflist}}

== References ==
* {{cite book |last1=James |first1=Robert C. |last2=James |first2=Glenn |year=1992 |title=Mathematics Dictionary |edition=Fifth |publisher=Chapman & Hall |isbn=9780412990410 |url=https://books.google.com/books?id=UyIfgBIwLMQC }}

== Further reading ==
* [[Hua Luogeng]] (1981) ''Starting with the unit circle'', [[Springer Verlag]], {{ISBN|0-387-90589-8}}.

== External links ==
* [https://www.youtube.com/watch?v=-ypicun4AbM&list=PL0F555888A4C2329B Homeomorphism and the Group Structure on a Circle]

[[Category:Lie groups]]