Circle group
508177
207829831
2008-04-24T11:10:40Z
92.12.7.4
/* Topological and analytic structure */ analytic function link
{{dablink|For the jazz group, see [[Circle (jazz band)]].}}
In [[mathematics]], the '''circle group''', denoted by '''T''' (or in [[blackboard bold]] by <math>\mathbb T</math>), is the multiplicative [[group (mathematics)|group]] of all [[complex number]]s with absolute value 1, i.e., the [[unit circle]] in the complex plane.
:<math>\mathbb T = \{ z \in \mathbb C : |z| = 1 \}.</math>
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 notation '''T''' for the circle group stems from the fact that '''T'''<sup>''n''</sup> (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 group]]s.
== Elementary introduction ==
<!-- this section is intended to be accessible to a curious high school student -->
[[Image:Circle-group.svg|thumb|200px|Addition on the circle group]]
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° − 360° = 60°.
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 (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).
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 1-dimensional [[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>.
==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
:<math>\mathbb T \cong \mbox{U}(1) \cong \mbox{SO}(2) \cong \mathbb R/\mathbb Z.\,</math>
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]].
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
:<math>\theta \mapsto e^{i\theta} = \cos\theta + i\sin\theta.</math>
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:
:<math>e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}.\,</math>
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
:<math>\mathbb T \cong \mathbb R/2\pi\mathbb Z.\,</math>
After rescaling we can also say that '''T''' is isomorphic to '''R'''/'''Z'''.
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
:<math> e^{i\theta} \leftrightarrow \begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}.</math>
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 [[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.
==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 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
:<math>\phi_n(e^{i\theta}) = e^{in\theta},\qquad n\in\mathbb Z.</math>
These representations are all inequivalent. The representation φ<sub>-n</sub> is [[conjugate representation|conjugate]] to φ<sub>n</sub>,
:<math>\phi_{-n} = \overline{\phi_n}.</math>
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>:
:<math>\mathrm{Hom}(\mathbb T,\mathbb T) \cong \mathbb Z.</math>
The irreducible [[real number|real]] representations of the circle group are the [[trivial representation]] (which is 1-dimensional) and the representations
:<math>\rho_n(e^{i\theta}) = \begin{bmatrix}
\cos n\theta & -\sin n\theta \\
\sin n\theta & \cos n\theta \\
\end{bmatrix},\quad n\in\mathbb Z^{+}.</math>
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>.
==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 ''n''th [[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
:<math>\mathbb T \cong \mathbb R \oplus (\mathbb Q / \mathbb Z).\,</math>
The isomorphism
:<math>\mathbb C^\times \cong \mathbb R \oplus (\mathbb Q / \mathbb Z)</math>
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'''.
==See also==
*[[Rotation number]]
*[[Torus]]
*[[One-parameter subgroup]]
*[[Unitary group]]
*[[Orthogonal group]]
==References==
{{Unreferenced|date=May 2007}}
[[Category:Group theory]]
[[Category:Topological groups]]
[[Category:Lie groups]]
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[[it:Gruppo circolare]]
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