Editing Compact group (section)

===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.