Circle group

In mathematics, the circle group, denoted by formula_1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers
The circle group forms a subgroup of formula_3, the multiplicative group of all nonzero complex numbers. Since formula_3 is abelian, it follows that formula_1 is as well. The circle group is also the group formula_6 of 1×1 complex-valued unitary matrices; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle formula_7 of rotation by
This is the exponential map for the circle group.
The circle group plays a central role in Pontryagin duality, and in the theory of Lie groups.
The notation formula_1 for the circle group stems from the fact that, with the standard topology (see below), the circle group is a 1-torus. More generally formula_10 (the direct product of formula_1 with itself formula_12 times) is geometrically an formula_12-torus.
Elementary introduction.
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 , 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 ).
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 , 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 object. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on formula_3, 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 formula_3 (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 one-parameter group, an instance of a Lie group. In fact, up to isomorphism, it is the unique 1 dimensional compact, connected Lie group. Moreover, every formula_12 dimensional compact, connected, abelian Lie group is isomorphic to formula_10.
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
Note that the slash (/) denotes here quotient group.
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 formula_6, the first unitary group.
The exponential function gives rise to a group homomorphism formula_20 from the additive real numbers formula_21 to the circle group formula_1 via the map
The last equality is Euler's formula or the complex exponential. The real number θ corresponds to the angle (in radians) on the unit circle as measured counterclockwise 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:
This exponential map is clearly a surjective function from formula_21 to formula_1. It is not, however, injective. The kernel of this map is the set of all integer multiples of formula_27. By the first isomorphism theorem we then have that
After rescaling we can also say that formula_1 is isomorphic to formula_30.
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
This function shows that the circle group is isomorphic to the special orthogonal group formula_32 since
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 formula_35 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 the formula_12th roots of unity form a cyclic group of which is unique up to isomorphism.
In the same way that the real numbers are a completion of the "b"-adic rationals for every natural number formula_37, the circle group is the completion of the Prüfer group for formula_38, given by the inverse limit formula_39.
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
must take values in formula_41. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself.
These representations are all inequivalent. The representation formula_42 is conjugate to formula_43,
These representations are just the characters of the circle group. The character group of formula_1 is clearly an infinite cyclic group generated by formula_46:
The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representations
taking values in formula_49. Here we only have positive integers formula_12 since the representation formula_51 is equivalent to formula_52.
Group structure.
The circle group formula_1 is a divisible group. Its torsion subgroup is given by the set of all formula_12th roots of unity for all formula_12, and is isomorphic to formula_56. The structure theorem for divisible groups and the axiom of choice together tell us that formula_1 is isomorphic to the direct sum of formula_58 with a number of copies of formula_59.
The number of copies of formula_60 must be formula_61 (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of formula_61 copies of formula_60 is isomorphic to formula_21, as formula_21 is a vector space of dimension formula_61 over formula_59. Thus
The isomorphism
can be proved in the same way, since formula_3 is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of formula_71.