Circle group

In mathematics, the circle group, denoted by

⁠, the multiplicative group of all nonzero complex numbers.

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 ⁠

The circle group plays a central role in Pontryagin duality and in the theory of Lie groups.

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.

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.

Moreover, since the unit circle is a closed subset of the complex plane, the circle group is a closed subgroup of

In fact, up to isomorphism, it is the unique 1-dimensional compact, connected Lie group.

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

The circle group shows up in a variety of forms in mathematics.

⁠ unitary matrices coincides with the circle group; the unitary condition is equivalent to the condition that its element have absolute value 1.

The exponential function gives rise to a map

corresponds to the angle (in radians) on the unit circle as measured counterclockwise from the positive x-axis.

The kernel of this map is the set of all integer multiples of ⁠

The unit complex numbers can be realized as 2×2 real orthogonal matrices, i.e.,

associating the squared modulus and complex conjugate with the determinant and transpose, respectively, of the corresponding matrix.

As the angle sum trigonometric identities imply that

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.

of dimension > 0 has 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.

The circle group has many subgroups, but its only proper closed subgroups consist of roots of unity: For each integer ⁠

th roots of unity form a cyclic group of order ⁠

In the same way that the real numbers are a completion of the b-adic rationals

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

The structure theorem for divisible groups and the axiom of choice together tell us that