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Compact group
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==Representation theory== The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) was founded by the [[Peter–Weyl theorem]].<ref>{{citation|first1=F.|last1=Peter|first2=H.|last2=Weyl|title=Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe|journal=Math. Ann.|volume=97|year=1927|pages=737–755|doi=10.1007/BF01447892}}.</ref> [[Hermann Weyl]] went on to give the detailed [[character theory]] of the compact connected Lie groups, based on [[maximal torus]] theory.<ref>{{harvnb|Hall|2015}} Part III</ref> The resulting [[Weyl character formula]] was one of the influential results of twentieth century mathematics. The combination of the Peter–Weyl theorem and the Weyl character formula led Weyl to a complete classification of the representations of a connected compact Lie group; this theory is described in the next section. A combination of Weyl's work and [[Closed-subgroup theorem|Cartan's theorem]] gives a survey of the whole representation theory of compact groups ''G''. That is, by the Peter–Weyl theorem the irreducible [[unitary representation]]s ρ of ''G'' are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If ''G'' is not itself a Lie group, there must be a kernel to ρ. Further one can form an [[inverse system]], for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies ''G'' as an [[inverse limit]] of compact Lie groups. Here the fact that in the limit a [[faithful representation]] of ''G'' is found is another consequence of the Peter–Weyl theorem. The unknown part of the representation theory of compact groups is thereby, roughly speaking, thrown back onto the [[complex representations of finite groups]]. This theory is rather rich in detail, but is qualitatively well understood.
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