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| {{Lie groups |Other}}
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| In [[mathematics]], the '''circle group''', denoted by '''T''', is the multiplicative [[group (mathematics)|group]] of all [[complex number]]s with [[Absolute_value#Complex_numbers|absolute value]] 1, i.e., the [[unit circle]] in the [[complex plane]].
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| :<math>\mathbb T = \{ z \in \mathbb C : |z| = 1 \}.</math>
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| The circle group forms a [[subgroup]] of '''C'''<sup>×</sup>, the multiplicative group of all nonzero complex numbers. Since '''C'''<sup>×</sup> is [[abelian group|abelian]], it follows that '''T''' is as well. The circle group is also the group '''U(1)''' of 1×1 [[unitary matrix|unitary matrices]]; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle θ of rotation by
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| :<math>\theta\mapsto z = e^{i\theta} = \cos\theta + i\sin\theta.</math>
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| This is the [[exponential map]] for the circle group.
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| The circle group plays a central role in [[Pontryagin duality]], and in the theory of [[Lie group]]s.
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| The notation '''T''' for the circle group stems from the fact that '''T'''<sup>''n''</sup> (the [[direct product of groups|direct product]] of '''T''' with itself ''n'' times) is geometrically an ''n''-[[torus]]. The circle group is then a 1-torus.
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| == Elementary introduction ==
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| <!-- this section is intended to be accessible to a curious high school student -->
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| [[Image:Circle-group.svg|thumb|200px|Multiplication on the circle group is equivalent to addition of angles]]
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| One way to think about the circle group is that it describes how to add ''angles'', where only angles between 0° and 360° are permitted. For example, the diagram illustrates how to add 150° to 270°. The answer should be 150° + 270° = 420°, 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 360° which gives 420° = 60° ([[Modular arithmetic|mod]] 360°).
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| Another description is in terms of ordinary addition, where only numbers between 0 and 1 are allowed (with 1 corresponding to a full rotation). 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.
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| ==Topological and analytic structure==
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| 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 '''C'''<sup>×</sup>, 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'''<sup>×</sup> (itself regarded as a topological group).
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| 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 ''n''-dimensional compact, connected, abelian Lie group is isomorphic to '''T'''<sup>''n''</sup>.
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| ==Isomorphisms==
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| 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
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| :<math>\mathbb T \cong \mbox{U}(1) \cong \mathbb R/\mathbb Z \cong \mbox{SO}(2).</math>
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| Note that the slash (/) denotes here [[quotient group]].
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| The set of all 1×1 [[unitary matrix|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]].
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| The [[exponential function]] gives rise to a [[group homomorphism]] exp : '''R''' → '''T''' from the additive real numbers '''R''' to the circle group '''T''' via the map
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| :<math>\theta \mapsto e^{i\theta} = \cos\theta + i\sin\theta.</math>
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| 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 that the multiplication of unit complex numbers corresponds to addition of angles:
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| :<math>e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}.\,</math>
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| This exponential map is clearly a [[surjective]] function from '''R''' to '''T'''. It is not, however, [[injective]]. The [[kernel (group theory)|kernel]] of this map is the set of all [[integer]] multiples of 2π. By the [[first isomorphism theorem]] we then have that
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| :<math>\mathbb T \cong \mathbb R/2\pi\mathbb Z.\,</math>
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| After rescaling we can also say that '''T''' is isomorphic to '''R'''/'''Z'''.
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| If complex numbers are realized as 2×2 real [[matrix (mathematics)|matrices]] (see [[complex number]]), the unit complex numbers correspond to 2×2 [[orthogonal matrices]] with unit [[determinant]]. Specifically, we have
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| :<math> e^{i\theta} \leftrightarrow \begin{bmatrix}
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| \cos \theta & -\sin \theta \\
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| \sin \theta & \cos \theta \\
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| \end{bmatrix}.</math>
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| 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.
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| ==Properties==
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| Every 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]].
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| 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 ''n'' > 0, the ''n''<sup>th</sup> roots of unity form a [[cyclic group]] of order ''n'', which is unique up to isomorphism.
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| ==Representations==
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| 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 ρ : '''T''' → ''GL''(1, '''C''') ≅ '''C'''<sup>×</sup>, 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
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| :<math>\phi_n(e^{i\theta}) = e^{in\theta},\qquad n\in\mathbb Z.</math>
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| These representations are all inequivalent. The representation ''φ''<sub>−''n''</sub> is [[conjugate representation|conjugate]] to ''φ''<sub>''n''</sub>,
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| :<math>\phi_{-n} = \overline{\phi_n}. \, </math>
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| These representations are just the [[character (mathematics)|characters]] of the circle group. The [[character group]] of '''T''' is clearly an [[infinite cyclic group]] generated by φ<sub>1</sub>:
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| :<math>\mathrm{Hom}(\mathbb T,\mathbb T) \cong \mathbb Z. \, </math> | |
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| The irreducible [[real number|real]] representations of the circle group are the [[trivial representation]] (which is 1-dimensional) and the representations
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| :<math>\rho_n(e^{i\theta}) = \begin{bmatrix}
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| \cos n\theta & -\sin n\theta \\
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| \sin n\theta & \cos n\theta
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| \end{bmatrix},\quad n\in\mathbb Z^{+},</math>
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| taking values in SO(2). Here we only have positive integers ''n'' since the representation <math>\rho_{-n}</math> is equivalent to <math>\rho_n</math>.
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| ==Group structure==
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| In this section we will forget about the topological structure of the circle group and look only at its structure as an abstract group.
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| The circle group '''T''' is a [[divisible group]]. Its [[torsion subgroup]] is given by the set of all ''n''th [[roots of unity]] for all ''n'', and is isomorphic to '''Q'''/'''Z'''. The [[Divisible group#Structure theorem of divisible groups|structure theorem]] for divisible groups tells us that '''T''' is isomorphic to the [[direct sum of abelian groups|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
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| :<math>\mathbb T \cong \mathbb R \oplus (\mathbb Q / \mathbb Z).\,</math>
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| The isomorphism
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| :<math>\mathbb C^\times \cong \mathbb R \oplus (\mathbb Q / \mathbb Z)</math>
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| can be proved in the same way, as '''C'''<sup>×</sup> is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of '''T'''.
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| ==See also==
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| {{Portal|Mathematics}}
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| *[[Rotation number]]
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| *[[Torus]]
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| *[[One-parameter subgroup]]
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| *[[Unitary group]]
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| *[[Orthogonal group]]
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| *[[Group of rational points on the unit circle]]
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| ==External links==
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| *[http://www.youtube.com/watch?v=-ypicun4AbM&list=PL0F555888A4C2329B Homeomorphism and the Group Structure on a Circle]
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| ==References==
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| * [[Hua Luogeng]] (1981) ''Starting with the unit circle'', [[Springer Verlag]], ISBN 0-387-90589-8 .
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| [[Category:Group theory]]
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| [[Category:Topological groups]]
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| [[Category:Lie groups]]
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