Editing Compact group (section)

==Compact Lie groups==

[[Lie group]]s form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include<ref>{{harvnb|Hall|2015}} Section 1.2</ref>
* the [[circle group]] '''T''' and the [[torus group]]s '''T'''<sup>''n''</sup>,
* the [[orthogonal group]] O(''n''), the [[special orthogonal group]] SO(''n'') and its covering [[spin group]] Spin(''n''),
* the [[unitary group]] U(''n'') and the [[special unitary group]] SU(''n''),
* the compact forms of the [[exceptional Lie group]]s: [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]].
The [[classification theorem]] of compact Lie groups states that up to finite [[group extension|extensions]] and finite [[covering group|covers]] this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.

===Classification===

Given any compact Lie group ''G'' one can take its [[identity component]] ''G''<sub>0</sub>, which is [[connected space|connected]]. The [[quotient group]] ''G''/''G''<sub>0</sub> is the group of components π<sub>0</sub>(''G'') which must be finite since ''G'' is compact. We therefore have a finite extension
:<math>1\to G_0 \to G \to \pi_0(G) \to 1.</math>
Meanwhile, for connected compact Lie groups, we have the following result:<ref>{{harvnb|Bröcker|tom Dieck|1985|loc=Chapter V, Sections 7 and 8}}</ref>
:'''Theorem''': Every connected compact Lie group is the quotient by a finite central subgroup of a product of a simply connected compact Lie group and a torus.
Thus, the classification of connected compact Lie groups can in principle be reduced to knowledge of the simply connected compact Lie groups together with information about their centers. (For information about the center, see the section below on fundamental group and center.)

Finally, every compact, connected, simply-connected Lie group ''K'' is a product of finitely many compact, connected, simply-connected [[simple Lie group]]s ''K''<sub>''i''</sub> each of which is isomorphic to exactly one of the following:
*The [[Symplectic group#Sp.28n.29|compact symplectic group]] <math>\operatorname{Sp}(n),\,n\geq 1</math>
*The [[special unitary group]] <math>\operatorname{SU}(n),\,n\geq 3</math>
*The [[spin group]] <math>\operatorname{Spin}(n),\,n\geq 7</math>
or one of the five exceptional groups [[G2 (mathematics)|G<sub>2</sub>]], [[F4 (mathematics)|F<sub>4</sub>]], [[E6 (mathematics)|E<sub>6</sub>]], [[E7 (mathematics)|E<sub>7</sub>]], and [[E8 (mathematics)|E<sub>8</sub>]]. The restrictions on ''n'' are to avoid special isomorphisms among the various families for small values of ''n''. For each of these groups, the center is known explicitly. The classification is through the associated [[root system]] (for a fixed maximal torus), which in turn are classified by their [[Dynkin diagram]]s.

The classification of compact, simply connected Lie groups is the same as the classification of complex [[semisimple Lie algebra]]s. Indeed, if ''K'' is a simply connected compact Lie group, then the complexification of the Lie algebra of ''K'' is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact, simply connected Lie group.

===Maximal tori and root systems===
{{See also|Maximal torus|Root system}}
A key idea in the study of a connected compact Lie group ''K'' is the concept of a ''maximal torus'', that is a subgroup ''T'' of ''K'' that is isomorphic to a product of several copies of <math>S^1</math> and that is not contained in any larger subgroup of this type. A basic example is the case <math>K = \operatorname{SU}(n)</math>, in which case we may take <math>T</math> to be the group of diagonal elements in <math>K</math>. A basic result is the ''torus theorem'' which states that every element of <math>K</math> belongs to a maximal torus and that all maximal tori are conjugate.

The maximal torus in a compact group plays a role analogous to that of the [[Semisimple Lie algebra#Cartan subalgebras and root systems|Cartan subalgebra]] in a complex semisimple Lie algebra. In particular, once a maximal torus <math>T\subset K</math> has been chosen, one can define a [[root system]] and a [[Weyl group]] similar to what one has for [[semisimple Lie algebra]]s.<ref>{{harvnb|Hall|2015}} Chapter 11</ref> These structures then play an essential role both in the classification of connected compact groups (described above) and in the representation theory of a fixed such group (described below).

The root systems associated to the simple compact groups appearing in the classification of simply connected compact groups are as follows:<ref>{{harvnb|Hall|2015}} Section 7.7</ref>
*The special unitary groups <math>\operatorname{SU}(n)</math> correspond to the root system <math>A_{n-1}</math>
*The odd spin groups <math>\operatorname{Spin}(2n+1)</math> correspond to the root system <math>B_{n}</math>
*The compact symplectic groups <math>\operatorname{Sp}(n)</math> correspond to the root system <math>C_{n}</math>
*The even spin groups <math>\operatorname{Spin}(2n)</math> correspond to the root system <math>D_{n}</math>
*The exceptional compact Lie groups correspond to the five exceptional root systems G<sub>2</sub>, F<sub>4</sub>, E<sub>6</sub>, E<sub>7</sub>, or E<sub>8</sub>

===Fundamental group and center===
{{See also|Fundamental group#Lie groups}}
It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its [[fundamental group]]. For compact Lie groups, there are [[Fundamental group#Lie groups|two basic approaches]] to computing the fundamental group. The first approach applies to the classical compact groups <math>\operatorname{SU}(n)</math>, <math>\operatorname{U}(n)</math>, <math>\operatorname{SO}(n)</math>, and <math>\operatorname{Sp}(n)</math> and proceeds by induction on <math>n</math>. The second approach uses the root system and applies to all connected compact Lie groups.

It is also important to know the center of a connected compact Lie group. The center of a classical group <math>G</math> can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in <math>G</math>. (The group SO(2) is an exception—the center is the whole group, even though most elements are not roots of the identity.) Thus, for example, the center of <math>\operatorname{SU}(n)</math> consists of ''n''th roots of unity times the identity, a cyclic group of order <math>n</math>.

In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus.<ref>{{harvnb|Hall|2015}} Section 13.8</ref> The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system <math>G_2</math> has trivial center. Thus, [[G2 (mathematics)|the compact <math>G_2</math> group]] is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are [[F4 (mathematics)|<math>F_4</math>]] and [[E8 (mathematics)|<math>E_8</math>]].)