In mathematics, the circle group, denoted by
T
The circle group forms a subgroup of, the multiplicative group of all nonzero complex numbers. Since
C x
T
A unit complex number in the circle group represents a rotation of the complex plane about the origin and can be parametrized by the angle measure :
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
T
Tn
T
n
n
The circle group is isomorphic to the special orthogonal group .
One way to think about the circle group is that it describes how to add angles, where only angles between 0° and 360° or
\in[0,2\pi)
\in(-\pi,+\pi]
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), i.e. the real numbers modulo the integers: . This can be achieved by throwing away the digits occurring before the decimal point. For example, when we work out the answer is 1.1666..., but we may throw away the leading 1, so the answer (in the circle group) is just
0.1\bar{6}\equiv1.1\bar{6}\equiv-0.8\bar{3} (mod\Z)
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, 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 x
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
n
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, the first unitary group.
\exp:R\toT
R
T
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
R
After rescaling we can also say that
T
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
SO(2)
x
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.
Every compact Lie group
G
The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer, the
n
\Z[\tfrac1b]
\Z[\tfrac1b]/\Z
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 representationmust take values in . Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself.
For each integer
n
\phin
\phi-n
These representations are just the characters of the circle group. The character group of
T
The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representationstaking values in . Here we only have positive integers, since the representation
\rho-n
The circle group
T
n
n
T
Q/Z
The number of copies of
Q
akc
akc
Q
R
akc
The isomorphismcan be proved in the same way, since
C x