Circle group

For the jazz group, see Circle (jazz band).

Lie groups
Classical groups General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)
Simple Lie groups List of simple Lie groups Infinite simple Lie groups: An, Bn, Cn, Dn, Exceptional simple Lie groups: G2 F4 E6 E7 E8
Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry
Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Representation of a Lie group Representation of a Lie algebra
Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group
v · mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, i.e., the unit circle in the complex plane. $\mathbb T = \{ z \in \mathbb C : |z| = 1 \}.$ The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers. Since C× is abelian, it follows that T is as well. The circle group is also the group U(1) of 1×1 unitary matrices; these act on the complex plane by rotation about the origin. The circle group can be parametrized by the angle θ of rotation by $\theta\mapsto z = e^{i\theta} = \cos\theta + i\sin\theta.$ 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 for the circle group stems from the fact that Tn (the direct product of T with itself n times) is geometrically an n-torus. The circle group is then a 1-torus. Elementary introduction Multiplication on the circle group is equivalent to addition of angles 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 150° + 270° = 420°, 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 420° = 60° (mod 360°). 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 0.784 + 0.925 + 0.446, 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 C×, 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× (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 n-dimensional compact, connected, abelian Lie group is isomorphic to Tn. 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 $\mathbb T \cong \mbox{U}(1) \cong \mathbb R/\mathbb Z \cong \mbox{SO}(2).$ 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 U(1), the first unitary group. The exponential function gives rise to a group homomorphism exp : R → T from the additive real numbers R to the circle group T via the map $\theta \mapsto e^{i\theta} = \cos\theta + i\sin\theta.$ The last equality is Euler's formula. The real number θ corresponds to the angle on the unit circle as measured 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: $e^{i\theta_1}e^{i\theta_2} = e^{i(\theta_1+\theta_2)}.\,$ This exponential map is clearly a surjective function from R to T. It is not, however, injective. The kernel of this map is the set of all integer multiples of 2π. By the first isomorphism theorem we then have that $\mathbb T \cong \mathbb R/2\pi\mathbb Z.\,$ After rescaling we can also say that T is isomorphic to R/Z. 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 $e^{i\theta} \leftrightarrow \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}.$ The circle group is therefore isomorphic to the special orthogonal group SO(2). This has the geometric interpretation that multiplication by a unit complex number is a proper rotation in the complex plane, and every such rotation is of this form. Properties Every compact Lie group G 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 n > 0, the nth roots of unity form a cyclic group of order n, which is unique up to isomorphism. 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 ρ : T → GL(1, C) ≅ C×, must take values in U(1) ≅ T. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself. Every such homomorphism is of the form $\phi_n(e^{i\theta}) = e^{in\theta},\qquad n\in\mathbb Z.$ These representations are all inequivalent. The representation φ−n is conjugate to φn, $\phi_{-n} = \overline{\phi_n}. \,$ These representations are just the characters of the circle group. The character group of T is clearly an infinite cyclic group generated by φ1: $\mathrm{Hom}(\mathbb T,\mathbb T) \cong \mathbb Z. \,$ The irreducible real representations of the circle group are the trivial representation (which is 1-dimensional) and the representations $\rho_n(e^{i\theta}) = \begin{bmatrix} \cos n\theta & -\sin n\theta \\ \sin n\theta & \cos n\theta \end{bmatrix},\quad n\in\mathbb Z^{+},$ taking values in SO(2). Here we only have positive integers n since the representation ρ − n is equivalent to ρn. Group structure In this section we will forget about the topological structure of the circle group and look only at its structure as an abstract group. The circle group T is a divisible group. Its torsion subgroup is given by the set of all nth roots of unity for all n, and is isomorphic to Q/Z. The structure theorem for divisible groups tells us that T is isomorphic to the direct sum of Q/Z with a number of copies of Q. The number of copies of Q must be c (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of c copies of Q is isomorphic to R, as R is a vector space of dimension c over Q. Thus $\mathbb T \cong \mathbb R \oplus (\mathbb Q / \mathbb Z).\,$ The isomorphism $\mathbb C^\times \cong \mathbb R \oplus (\mathbb Q / \mathbb Z)$ can be proved in the same way, as C× is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of T. See also Mathematics portal Rotation number Torus One-parameter subgroup Unitary group Orthogonal group Group of rational points on the unit circle References Hua Luogeng (1981) Starting with the unit circle Springer Verlag. Wikimedia Foundation. 2010. Игры ⚽ Поможем сделать НИР Kyphosis Caddo Look at other dictionaries: circle of friends — group of pals, social circle, group of friends …   English contemporary dictionary Circle — This article is about the shape and mathematical concept. For other uses, see Circle (disambiguation). Circle illustration showing a radius, a diameter, the centre and the circumference …   Wikipedia circle — n. circular geometric figure 1) to describe, draw a circle 2) to square a circle (to square a circle is impossible) 3) the Antarctic; Arctic; polar circle (at the Arctic circle) 4) the great circle (on the earth s surface) 5) a traffic circle… …   Combinatory dictionary circle — Synonyms and related words: O, acquaintance, advance, alentours, alternate, ambience, ambit, anklet, annular muscle, annulus, anthelion, antisun, aphelion, apogee, arc, arena, areola, armlet, arsis, ascend, associates, astronomical longitude,… …   Moby Thesaurus Circle time — Circle time, also called group time, refers to any time that a group of people are sitting together for an activity involving everyone. The method is now in widespread use in schools across the UK. In Scotland many primary schools use the method… …   Wikipedia Circle Time — Circle Time, also called group time, refers to any time that a group of people are sitting together for an activity involving everyone. The method is now in widespread use in schools across the UK. In Scotland many primary schools use the method… …   Wikipedia Circle Housing Group — Circle is one of the largest groups of housing associations in the UK. It provides affordable housing and related services in England, mainly in London, the South East, East Anglia and Birmingham. The group was formed as Circle Anglia when Circle …   Wikipedia Circle Jerks discography — Circle Jerks discography Releases ↙Studio albums 6 ↙Live albums 1 …   Wikipedia Circle Bakote — is a cluster of Union Councils in the eastern part of Abbottabad District in the North West Frontier Province of Pakistan. Circle Bakote is located on the upper and west bank of the Jhelum River at Kohalla Bridge. The region is somewhere between… …   Wikipedia Circle Jerks — Le chanteur de Circle Jerks, Keith Morris, lors d un concert à Austin, Texas, le 1er décembre 2006 Pays d’origine …   Wikipédia en Français 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”