In mathematics, the circle group, denoted by or , 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 plays a fundamental role in many areas of mathematics.
When a given unit complex number multiplies the other points of the circle, the effect is to rotate them through an angle determined by . In this way, the circle group becomes the group of symmetries of the circle which preserve its orientation (do not flip it). Composition of two rotations is the ordinary multiplication of complex numbers. Multiplication is commutative, , making the circle group commutative (an abelian group); correspondingly several rotations of the plane can be composed in any order with the same result. Rotations can alternately be parametrized by the angle measure , which is related to by the complex exponential function:
The circle group is sometimes denoted by , which is the unitary group of complex matrices. It is structurally the same as (i.e., isomorphic to) the group of 2-dimensional rotation matrices, i.e., the special orthogonal group . The angle measure gives a periodic parameterization of the circle group, so the group is often treated as a periodic interval with the endpoints glued together, and the group operation as addition modulo : the usual addition of angles. The notation for the circle group stems from the fact that a circle is a 1-dimensional torus. More generally, (the direct product of with itself times) is geometrically an -torus. The notation stands for 1-dimensional sphere.
The circle group has applications throughout mathematics, especially in advanced mathematics. It is the group underlying classical Fourier series. It is dual to the additive group of the integers. It also has applications throughout topology and mathematical physics. It is the group underlying electromagnetism. Electromagnetic theory can be formulated as a theory of bundles associated to the circle group, which are primary objects of study in homotopy theory and algebraic topology.
Elementary introduction
Elements of the circle group can be thought of as representing rotations of the Euclidean plane about a particular point, those geometric transformations which keep that point fixed while preserving the distance between arbitrary points as well as the orientation of shapes. The group operation then corresponds to composition of rotations, that is, the combination of two rotations resulting from applying one rotation and then the other.
If a point with rectangular coordinates in the Euclidean plane is represented by a complex number (see complex plane), then multiplication by a unit complex number results in the complex number which represents the point , the result of rotating around the origin by some rotation . Each unit complex number corresponds to a different rotation, and vice versa. So the unit complex number can be used to represent the corresponding rotation . If two different rotations and are represented by the unit complex numbers and , then their composition is represented by the product of the complex numbers .
Another common way of representing a rotation is by an angle measured in degrees or radians, often denoted by the Greek letter . Conventionally, a positive angle is considered to represent a counterclockwise rotation. Two angles which differ by or radians represent the same rotation, so it is common to pick the representative angle for the rotation to be between and (or radians) or, more symmetrically, between and (or radians). Two rotations represented by angle measure can be composed simply by adding them together; sometimes the new angle falls outside the designated interval for representative angles – we have looped around the circle – and we can then add or subtract a multiple of ( radians) to return to the interval.
A rotation of a point in the complex plane by an angle about the origin (measured in radians) corresponds to multiplication by the complex number , where is the complex exponential function and and are trigonometric functions). Adding angles corresponds to multiplying unit complex numbers:
In some contexts, instead of using degrees in the interval or radians in the interval , angles are represented by fractions of a full turn in the interval . This version makes arithmetic convenient because an angle which has fallen outside of the interval can be brought back inside it by taking its fractional part: for a positive number, this amounts to throwing away any digits before the decimal point. For example, to compose a turn rotation and a turn rotation, add the two numbers, , and then throw away the leading , resulting in a turn rotation.
The periodic interval can be considered to be the quotient group of the real numbers modulo the integers, symbolically written ; this is the group of equivalence classes of real numbers, with any two real numbers which differ by an integer considered to be equivalent. Each equivalence class includes one real number in the finite interval , which can be used to represent it. As a group under addition, is isomorphic to the circle group of unit complex numbers under multiplication, meaning the two groups are structurally the same.
Isomorphisms and conventions 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
where the slash () denotes group quotient and the existence of an isomorphism between the groups.
Unitary group U(1) A matrix has row and column, and only a single entry. The product of two matrices is the matrix whose single entry is the product of the entries of each of the matrices:
Therefore any group of matrices under matrix multiplication can be trivially related to the group of corresponding entries under multiplication. A unitary matrix is a matrix with complex entries for which multiplication by the conjugate transpose results in the identity matrix; for a matrix, an equivalent condition is that the single entry has unit absolute value, making it an element of the circle group. This makes the circle group canonically isomorphic to the first unitary group , the group of unitary matrices under matrix multiplication. That is,
It is routine to conflate a matrix with its single entry, considering elements of to be unit complex numbers directly. The symbol can be used as an abbreviation for the multiplicative group of unit complex numbers.
Periodic interval under addition The exponential function gives rise to a map from the additive real numbers to the circle group , elements of which can be separated into real and imaginary components using Euler's formula,
where the real number corresponds to an angle (argument) of a unit complex number as measured counterclockwise from the positive real axis. The exponential function converts sums in the domain to products in the image. In particular, for any pair of real numbers and ,
This property makes a group homomorphism. While the map is surjective, it is not injective and therefore not an isomorphism. The kernel of this map (the elements which map to the complex number ) is the set of all integer multiples of . By the first isomorphism theorem we then have that
After rescaling we can also say that is isomorphic to .
Special orthogonal group SO(2) Rotations of the Euclidean vector plane about the origin are examples of linear transformations, which can be represented as matrices with real entries, specifically as rotation matrices (also called special orthogonal matrices), those matrices which yield the identity matrix when multiplied by their transpose, , and whose determinant is 1. Such a matrix always takes the form
with . The matrix can be uniquely identified with the unit complex number . If the matrix represents a rotation by angle , then , , and , so can be written as
The squared modulus of the complex number, corresponds to the determinant and the complex conjugate, , corresponds to the transpose of the matrix. Matrix multiplication of such matrices is consistent with addition of corresponding angles or multiplication of corresponding complex numbers (see Angle sum identities).
Therefore the circle group is isomorphic to the special orthogonal group , that is,
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.
Conventions While the symbol and the name "circle group" often refer to the unit complex numbers under multiplication, it is common to give this name and symbol to one of the other isomorphic groups, depending on convenience. The three common definitions of the circle group are:
In the first, the group operation is written multiplicatively, while in the latter two the operation is written additively.
Topological, measurable, 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. It is a metric space with (at least) two natural notions of distance between elements: the absolute value of the difference between two unit complex numbers (the length of the chord between points in the complex plane), or the absolute value of the difference between nearest representative angles (the arc length of the shortest arc between points on the complex unit circle). The open balls are circular arcs. 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 (itself regarded as a topological group). It is compact, connected, but not simply connected. Its fundamental group is the additive group of integers, with each integer corresponding to the number of times a closed path loops around the circle.
Integration and Haar measure Integration in the circle group can be defined in a canonical invariant way. For a continuous function , the integral takes the form: For the convention ,
For the convention ,
For the convention ,
Here, and refer to the usual Lebesgue measure on the real line. In each case, this is a probability measure, meaning that the integral of the constant function over the whole group is equal to 1. This integral is not arbitrary, but is the unique (Borel) probability measure that is invariant under constant change of variables for (or for or ). The asserted uniqueness is a special case of Haar's theorem, and the measure is known as the (normalized) Haar measure.
Invariant metrics
The chordal and length metrics already defined give a pair of invariant metrics on the circle group. They are invariant in the sense that the distance between two points is unchanged if the points are rotated through the same angle. These metrics are equivalent, in the sense that there is a positive constant with
for every pair of points . The arclength metric is the only invariant Riemannian metric on the circle, and in fact the only invariant length metric, up to a constant normalization.
Other invariant distance functions defining an invariant metric space structure on the circle group can also be defined. In general, an invariant metric is determined by specifying the distance from a point to each point , . If is continuous, satisfies , for , , and , then defines a compatible invariant metric on .