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Wikipedia / Circle group

In mathematics, the circle group 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.

. 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

, which is the unitary group 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 with the endpoints glued together, and the group operation as addition modulo for the circle group stems from the fact that a circle is a 1-dimensional torus (the direct product-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

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

where the slash () denotes group quotient 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 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

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 from the additive real numbers to the circle group , elements of which can be separated into real and imaginary components using Euler's formula 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 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 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 with . The matrix can be uniquely identified with the unit complex number . If the matrix represents a rotation by angle , then , , and , so

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 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 (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

, the integral takes the form:

  • ,

    ,

    ,

refer to the usual Lebesgue measure on the real line. In each case, this is a probability measure over the whole group is equal to 1. This integral is not arbitrary, but is the unique (Borel). 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. 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.

, called a snowflake metric. With this metric, the Hausdorff dimension, and so gives the group the structure of a fractal curve can be embedded in the plane into the Koch snowflake plane curve, whose chordal metric is equivalent.

Other invariant metrics on the circle are used in areas of data analysis and machine learning, and exhibit similar metric-measure properties as the snowflake metrics. One family derives from the study of diffusion on the circle, or equivalently is given by the Sobolev distance between Dirac delta functions on the circle group:

Manifold and Lie group structure

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-dimensional compact, connected, abelian Lie group is isomorphic to .

of positive dimension contains a non-trivial maximal torus as a subgroup, and therefore a subgroup isomorphic to the circle group. This 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.

Subgroups

The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer , there are th roots of unity, which form a cyclic group of order, which is the unique subgroup of that order.

All other subgroups of the circle group have infinitely many elements and are dense. Some examples are:

Endomorphisms and covering maps

The continuous group endomorphisms-fold covering homomorphismth roots of unity. In particular, the automorphism group.

Representations

The finite-dimensional continuous representations of the circle group are easy to describe. It follows from Schur's lemma that the irreducible finite-dimensional continuous complex must take values in . Therefore, the irreducible finite-dimensional continuous complex representations of the circle group are just the continuous homomorphisms from the circle group to itself.

of the circle group by is conjugate to

These representations are just the continuous characters of the circle group. The continuous character group, also called its Pontryagin dual, is an infinite cyclic group generated by

The finite-dimensional continuous irreducible real representations of the circle group are the trivial representation taking values in . Here we only have positive integers is equivalent to .

The representations of the circle group play a role in constructing the representations of all compact Lie groups. Every compact Lie group has a maximal torus is called the rank of the Lie group. The representations of the torus determine the weight lattice, and the representations of the full group can be found by highest weight theory.

Fourier analysis

Fourier series

are the fundamental building blocks of Fourier series. The characters form an orthonormal basisth Fourier coefficient.

In the three common conventions for realizing the circle group, the characters are:

  • .

  • .

  • .

in different parts of the notation. The first is the unit-circle or Laurent convention, it is closely related to the bilateral z-transform. The second is the angular frequency appearing in the character.

Heat equation and diffusion

Diffusion is an invariant process on the circle that governs the statistics of motion of Brownian particles and the flow of heat in a one-dimensional periodic domain. Diffusion is governed by the standard translation-invariant Laplacian Thus the heat kernel Equivalently, by Poisson summation This is a periodized Gaussian and may be expressed in terms of a Jacobi theta function is convolution.

, can be used to study the Schrödinger equation.

Use in central extensions

The circle group serves an important role in projective representations of other groups. A projective unitary representation to the projective unitary group of a Hilbert space that makes a choice of phase factor is a 2-cocycle for which the representation becomes linear rather than projective, by taking the central extension. This construction underlies the Heisenberg group, where phase factors give the central extension appearing in the canonical commutation relations and the Stone–von Neumann theorem. In conformal field theory, one version of the Virasoro group is the universal central extension of the oriented diffeomorphisms of the circle by the circle group.

Ergodic theory

The circle group is a fundamental starting point in the study of ergodic theory and dynamical systems. Several natural questions are:

  • What structure does the orbit of a point have? For instance, is it dense in the group? Is the orbit of a point equidistributed?

  • ?

or not.

, and so the orbits are periodic. If not, then the orbit is dense, and in fact equidistributed.

is minimal and acts ergodically is irrational. It is uniquely ergodic in that case. One version More formally, for almost every.

Line bundles and gauge theory

The circle group appears naturally in the theory of Hermitian complex line bundles over manifolds and algebraic varieties such as Riemann surfaces is a complex line bundle with a Hermitian metric, the unit vectors in the fibers from a U(1)-bundle-bundle determines an associated complex line bundle.

-bundles are classified by the first Chern class. Equivalently, the classifying space, is an infinite complex projective space of rays in a separable complex Hilbert space, whose second cohomology group is generated by the first Chern class.

In gauge theory is the structure group of electromagnetism. A connection, which corresponds to the electromagnetic potential, whose curvature two-form represents the field.

Pontryagin duality and Banach algebras

The Pontryagin dual.

These dualities are reflected in the associated commutative Banach algebras. The group algebra, with convolution, has Gelfand transform; equivalently its space of maximal ideals. The image is the Wiener algebra, the algebra of continuous functions on the circle with absolutely convergent Fourier series..

The same duality appears for group C^*-algebras. The statement encodes the Riemann-Lebesgue lemma for Fourier series.

The circle group also appears as the basic building block of solenoids, the inverse limit is a compact connected abelian group. Its Pontryagin dual is the direct limit. In this way solenoids are one-dimensional compact connected abelian groups obtained from the circle by inverse limits of covering homomorphisms.

Every Pontryagin dual of a torsion-free rank one discrete abelian group is either a circle or a solenoid. give the one-dimensional solenoids.

Abstract group structure

can be regarded as an abstract group without regard for its topology, or equivalently as a (non-compact) topological group equipped with the discrete topology.

The circle group is a divisible group. Its torsion subgroup-th roots of unity and is isomorphic to . The structure theorem for divisible groups and the axiom of choice is isomorphic, as an abstract group, to the direct sum with a number of copies of .

The number of copies of (the cardinality of the continuum copies of is isomorphic to is a vector space over

can be proved in the same way, since is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of .

Algebraic group structure

The circle group can also be viewed as an algebraic torus. The field norm and the circle group is the kernel In algebraic group terminology, this is the group of real points of the norm-one torus associated with the field extension.

Non-standard analysis

In non-standard analysis, the most natural replacement of the circle group is the hyperfinite cyclic group is an infinite hyperinteger, that is a number that is greater than every standard natural number. The standard part. Moreover, it carries a natural normalized counting measure, whose associated Loeb measure (the pushforward measure.

is the natural setting for non-standard Fourier analysis. One reason is that, being a cyclic group, it is isomorphic to its dual group (internally). Non-standard Fourier series are ordinary (hyper-)finite sums, whose standard part recovers traditional Fourier series on the circle group.

, which is not hyperfinite, and so the Fourier theory on the full non-standard circle group is less finite.

See also

Notes

References