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Compact group

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about mathematics. For astronomy of galaxies, see <a href="Galaxy_group" title="Galaxy group">galaxy group</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a compact (topological) group is a <a href="Topological_group" title="Topological group">topological group</a> whose <a href="Topology" title="Topology">topology</a> realizes it as a <a href="Compact_space" title="Compact space">compact topological space</a> (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of <a href="Finite_group" title="Finite group">finite groups</a> with the <a href="Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a> and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group actions</a> and <a href="Representation_theory" title="Representation theory">representation theory</a>.

In the following we will assume all groups are <a href="Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a>.

<div class="mw-heading mw-heading2"><h2 id="Compact_Lie_groups">Compact Lie groups</h2></div>

<a href="Lie_group" title="Lie group">Lie groups</a> form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups include<a href="#cite_note-1">[1]</a>

<ul><li>the <a href="Circle_group" title="Circle group">circle group</a> T and the <a href="Torus_group" class="mw-redirect" title="Torus group">torus groups</a> Tn,</li>

<li>the <a href="Orthogonal_group" title="Orthogonal group">orthogonal group</a> O(n), the <a href="Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">special orthogonal group</a> SO(n) and its covering <a href="Spin_group" title="Spin group">spin group</a> Spin(n),</li>

<li>the <a href="Unitary_group" title="Unitary group">unitary group</a> U(n) and the <a href="Special_unitary_group" title="Special unitary group">special unitary group</a> SU(n),</li>

<li>the compact forms of the <a href="Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">exceptional Lie groups</a>: <a href="G2_(mathematics)" title="G2 (mathematics)">G2</a>, <a href="F4_(mathematics)" title="F4 (mathematics)">F4</a>, <a href="E6_(mathematics)" title="E6 (mathematics)">E6</a>, <a href="E7_(mathematics)" title="E7 (mathematics)">E7</a>, and <a href="E8_(mathematics)" title="E8 (mathematics)">E8</a>.</li></ul>

The <a href="Classification_theorem" title="Classification theorem">classification theorem</a> of compact Lie groups states that up to finite <a href="Group_extension" title="Group extension">extensions</a> and finite <a href="Covering_group" title="Covering group">covers</a> this exhausts the list of examples (which already includes some redundancies). This classification is described in more detail in the next subsection.

<div class="mw-heading mw-heading3"><h3 id="Classification">Classification</h3></div>

Given any compact Lie group G one can take its <a href="Identity_component" title="Identity component">identity component</a> G0, which is <a href="Connected_space" title="Connected space">connected</a>. The <a href="Quotient_group" title="Quotient group">quotient group</a> G/G0 is the group of components π0(G) which must be finite since G is compact. We therefore have a finite extension

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<annotation encoding="application/x-tex">{\displaystyle 1\to G_{0}\to G\to \pi _{0}(G)\to 1.}</annotation>

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</math><img src="./6556eac77b6c4e7b94c2a537a3f995580325be05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.151ex; height:2.843ex;" alt="{\displaystyle 1\to G_{0}\to G\to \pi _{0}(G)\to 1.}" loading="lazy"></dd></dl>

Meanwhile, for connected compact Lie groups, we have the following result:<a href="#cite_note-2">[2]</a>

<dl><dd>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.</dd></dl>

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 <a href="Simple_Lie_group" title="Simple Lie group">simple Lie groups</a> Ki each of which is isomorphic to exactly one of the following:

<ul><li>The <a href="Symplectic_group#Sp.28n.29" title="Symplectic group">compact symplectic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (n),\,n\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (n),\,n\geq 1}</annotation>

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</math><img src="./0e9a0d230ef045b0db99837b1cc797909bc03fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.866ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (n),\,n\geq 1}" loading="lazy"></li>

<li>The <a href="Special_unitary_group" title="Special unitary group">special unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SU} (n),\,n\geq 3}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SU} (n),\,n\geq 3}</annotation>

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</math><img src="./bb4967ee89614a40c6c0567c03c7acfdecd38c31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.316ex; height:2.843ex;" alt="{\displaystyle \operatorname {SU} (n),\,n\geq 3}" loading="lazy"></li>

<li>The <a href="Spin_group" title="Spin group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (n),\,n\geq 7}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (n),\,n\geq 7}</annotation>

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</math><img src="./445bbd2e3fc423af76b85f0a228a4ab51511897c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.805ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (n),\,n\geq 7}" loading="lazy"></li></ul>

or one of the five exceptional groups <a href="G2_(mathematics)" title="G2 (mathematics)">G2</a>, <a href="F4_(mathematics)" title="F4 (mathematics)">F4</a>, <a href="E6_(mathematics)" title="E6 (mathematics)">E6</a>, <a href="E7_(mathematics)" title="E7 (mathematics)">E7</a>, and <a href="E8_(mathematics)" title="E8 (mathematics)">E8</a>. 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 <a href="Root_system" title="Root system">root system</a> (for a fixed maximal torus), which in turn are classified by their <a href="Dynkin_diagram" title="Dynkin diagram">Dynkin diagrams</a>.

The classification of compact, simply connected Lie groups is the same as the classification of complex <a href="Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebras</a>. 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.

<div class="mw-heading mw-heading3"><h3 id="Maximal_tori_and_root_systems">Maximal tori and root systems</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Maximal_torus" title="Maximal torus">Maximal torus</a> and <a href="Root_system" title="Root system">Root system</a></div>

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

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<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>

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</math><img src="./60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"> and that is not contained in any larger subgroup of this type. A basic example is the case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\operatorname {SU} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle K=\operatorname {SU} (n)}</annotation>

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</math><img src="./bffed2439ea4c0cf13d5bce8b953957274723e06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.404ex; height:2.843ex;" alt="{\displaystyle K=\operatorname {SU} (n)}" loading="lazy">, in which case we may take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> to be the group of diagonal elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. A basic result is the torus theorem which states that every element of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> 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 <a href="Semisimple_Lie_algebra#Cartan_subalgebras_and_root_systems" title="Semisimple Lie algebra">Cartan subalgebra</a> in a complex semisimple Lie algebra. In particular, once a maximal torus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\subset K}">

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<annotation encoding="application/x-tex">{\displaystyle T\subset K}</annotation>

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</math><img src="./6ed49747cee06f73367c68ddf0bdad17bddcf258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.801ex; height:2.176ex;" alt="{\displaystyle T\subset K}" loading="lazy"> has been chosen, one can define a <a href="Root_system" title="Root system">root system</a> and a <a href="Weyl_group" title="Weyl group">Weyl group</a> similar to what one has for <a href="Semisimple_Lie_algebra" title="Semisimple Lie algebra">semisimple Lie algebras</a>.<a href="#cite_note-3">[3]</a> 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:<a href="#cite_note-4">[4]</a>

<ul><li>The special unitary groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SU} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SU} (n)}</annotation>

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</math><img src="./31937ff2e5d88617cd98e982a475a6bf481b321c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle \operatorname {SU} (n)}" loading="lazy"> correspond to the root system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n-1}}">

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<annotation encoding="application/x-tex">{\displaystyle A_{n-1}}</annotation>

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</math><img src="./b576e75f0336d126580faaad6039e2e84f6f3ee2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.062ex; height:2.509ex;" alt="{\displaystyle A_{n-1}}" loading="lazy"></li>

<li>The odd spin groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (2n+1)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (2n+1)}</annotation>

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</math><img src="./ee25a643bc794f5ce0eaad1eac27c5e3b44769da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.894ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (2n+1)}" loading="lazy"> correspond to the root system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation>

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</math><img src="./2f568bf6d34e97b9fdda0dc7e276d6c4501d2045.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}" loading="lazy"></li>

<li>The compact symplectic groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (n)}</annotation>

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</math><img src="./feb323e858a2a2d45958a3a2d052ff631dfc2964.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (n)}" loading="lazy"> correspond to the root system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation>

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</math><img src="./0301812adb392070d834ca2df4ed97f1cf132f33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}" loading="lazy"></li>

<li>The even spin groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spin} (2n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Spin} (2n)}</annotation>

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</math><img src="./d8bb79bfe0516a41ac01d2d58c49635a0a88f9e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.891ex; height:2.843ex;" alt="{\displaystyle \operatorname {Spin} (2n)}" loading="lazy"> correspond to the root system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle D_{n}}</annotation>

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</math><img src="./0fe03857347bf517e7fbda4085b0dafd6018cf18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle D_{n}}" loading="lazy"></li>

<li>The exceptional compact Lie groups correspond to the five exceptional root systems G2, F4, E6, E7, or E8</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Fundamental_group_and_center">Fundamental group and center</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Fundamental_group#Lie_groups" title="Fundamental group">Fundamental group § Lie groups</a></div>

It is important to know whether a connected compact Lie group is simply connected, and if not, to determine its <a href="Fundamental_group" title="Fundamental group">fundamental group</a>. For compact Lie groups, there are <a href="Fundamental_group#Lie_groups" title="Fundamental group">two basic approaches</a> to computing the fundamental group. The first approach applies to the classical compact groups <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SU} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SU} (n)}</annotation>

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</math><img src="./31937ff2e5d88617cd98e982a475a6bf481b321c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle \operatorname {SU} (n)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {U} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {U} (n)}</annotation>

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</math><img src="./1c87a50f9d99d0bf5a1506d6d37ccbd31d31e3d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.947ex; height:2.843ex;" alt="{\displaystyle \operatorname {U} (n)}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SO} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SO} (n)}</annotation>

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</math><img src="./79c49648527ab4aacf6c03c15633727606cc7d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.305ex; height:2.843ex;" alt="{\displaystyle \operatorname {SO} (n)}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (n)}</annotation>

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</math><img src="./feb323e858a2a2d45958a3a2d052ff631dfc2964.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (n)}" loading="lazy"> and proceeds by induction on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> can easily be computed "by hand," and in most cases consists simply of whatever roots of the identity are in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. (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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SU} (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {SU} (n)}</annotation>

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</math><img src="./31937ff2e5d88617cd98e982a475a6bf481b321c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.24ex; height:2.843ex;" alt="{\displaystyle \operatorname {SU} (n)}" loading="lazy"> consists of nth roots of unity times the identity, a <a href="Cyclic_group" title="Cyclic group">cyclic group</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

In general, the center can be expressed in terms of the root lattice and the kernel of the exponential map for the maximal torus.<a href="#cite_note-5">[5]</a> The general method shows, for example, that the simply connected compact group corresponding to the exceptional root system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"> has trivial center. Thus, <a href="G2_(mathematics)" title="G2 (mathematics)">the compact <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation>

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</math><img src="./645011b0c6933a02f5f7d84624f78220d747427e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}" loading="lazy"> group</a> is one of very few simple compact groups that are simultaneously simply connected and center free. (The others are <a href="F4_(mathematics)" title="F4 (mathematics)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{4}}</annotation>

</semantics>

</math><img src="./8718a2df1e70bea3cd21ab9e0cd45dc354818451.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{4}}" loading="lazy"></a> and <a href="E8_(mathematics)" title="E8 (mathematics)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{8}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{8}}</annotation>

</semantics>

</math><img src="./48479e96d90b4cfabc7784106cc3cfff907dda34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{8}}" loading="lazy"></a>.)

<div class="mw-heading mw-heading2"><h2 id="Further_examples">Further examples</h2></div>

Amongst groups that are not Lie groups, and so do not carry the structure of a <a href="Manifold" title="Manifold">manifold</a>, examples are the additive group Zp of <a href="P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a>, and constructions from it. In fact any <a href="Profinite_group" title="Profinite group">profinite group</a> is a compact group. This means that <a href="Galois_group" title="Galois group">Galois groups</a> are compact groups, a basic fact for the theory of <a href="Algebraic_extension" title="Algebraic extension">algebraic extensions</a> in the case of infinite degree.

<a href="Pontryagin_duality" title="Pontryagin duality">Pontryagin duality</a> provides a large supply of examples of compact commutative groups. These are in duality with abelian <a href="Discrete_group" title="Discrete group">discrete groups</a>.

<div class="mw-heading mw-heading2"><h2 id="Haar_measure">Haar measure</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a></div>

Compact groups all carry a <a href="Haar_measure" title="Haar measure">Haar measure</a>,<a href="#cite_note-6">[6]</a> which will be invariant by both left and right translation (the <a href="Haar_measure" title="Haar measure">modulus function</a> must be a continuous <a href="Homomorphism" title="Homomorphism">homomorphism</a> to <a href="Positive_reals" class="mw-redirect" title="Positive reals">positive reals</a> (R+, ×), and so 1). In other words, these groups are <a href="Unimodular_group" class="mw-redirect" title="Unimodular group">unimodular</a>. Haar measure is easily normalized to be a <a href="Probability_measure" title="Probability measure">probability measure</a>, analogous to dθ/2π on the circle.

Such a Haar measure is in many cases easy to compute; for example for orthogonal groups it was known to <a href="Adolf_Hurwitz" title="Adolf Hurwitz">Adolf Hurwitz</a>, and in the Lie group cases can always be given by an invariant <a href="Differential_form" title="Differential form">differential form</a>. In the profinite case there are many subgroups of <a href="Finite_index" class="mw-redirect" title="Finite index">finite index</a>, and Haar measure of a coset will be the reciprocal of the index. Therefore, integrals are often computable quite directly, a fact applied constantly in <a href="Number_theory" title="Number theory">number theory</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is a compact group and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is the associated Haar measure, the <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a> provides a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}(K,dm)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}(K,dm)}</annotation>

</semantics>

</math><img src="./00638f763e51282be5ec896580cfd67b5038ac21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.803ex; height:3.176ex;" alt="{\displaystyle L^{2}(K,dm)}" loading="lazy"> as an orthogonal direct sum of finite-dimensional subspaces of matrix entries for the irreducible representations of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Representation_theory">Representation theory</h2></div>

The representation theory of compact groups (not necessarily Lie groups and not necessarily connected) was founded by the <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a>.<a href="#cite_note-7">[7]</a> <a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a> went on to give the detailed <a href="Character_theory" title="Character theory">character theory</a> of the compact connected Lie groups, based on <a href="Maximal_torus" title="Maximal torus">maximal torus</a> theory.<a href="#cite_note-8">[8]</a> The resulting <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a> 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 <a href="Closed-subgroup_theorem" title="Closed-subgroup theorem">Cartan's theorem</a> gives a survey of the whole representation theory of compact groups G. That is, by the Peter–Weyl theorem the irreducible <a href="Unitary_representation" title="Unitary representation">unitary representations</a> 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 <a href="Inverse_system" class="mw-redirect" title="Inverse system">inverse system</a>, for the kernel of ρ smaller and smaller, of finite-dimensional unitary representations, which identifies G as an <a href="Inverse_limit" title="Inverse limit">inverse limit</a> of compact Lie groups. Here the fact that in the limit a <a href="Faithful_representation" title="Faithful representation">faithful representation</a> 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 <a href="Complex_representations_of_finite_groups" class="mw-redirect" title="Complex representations of finite groups">complex representations of finite groups</a>. This theory is rather rich in detail, but is qualitatively well understood.

<div class="mw-heading mw-heading2"><h2 id="Representation_theory_of_a_connected_compact_Lie_group">Representation theory of a connected compact Lie group</h2></div>

Certain simple examples of the representation theory of compact Lie groups can be worked out by hand, such as the representations of the <a href="Representation_of_a_Lie_group#An_example:_The_rotation_group_SO.283.29" title="Representation of a Lie group">rotation group SO(3)</a>, the <a href="Representation_theory_of_SU(2)" title="Representation theory of SU(2)">special unitary group SU(2)</a>, and the <a href="Clebsch%E2%80%93Gordan_coefficients_for_SU(3)#Representations_of_the_SU.283.29_group" title="Clebsch–Gordan coefficients for SU(3)">special unitary group SU(3)</a>. We focus here on the general theory. See also the parallel theory of <a href="Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras" title="Lie algebra representation">representations of a semisimple Lie algebra</a>.

Throughout this section, we fix a connected compact Lie group K and a <a href="Maximal_torus" title="Maximal torus">maximal torus</a> T in K.

<div class="mw-heading mw-heading3"><h3 id="Representation_theory_of_T">Representation theory of T</h3></div>

Since T is commutative, <a href="Schur's_lemma" title="Schur's lemma">Schur's lemma</a> tells us that each irreducible representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> of T is one-dimensional:

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}.}</annotation>

</semantics>

</math><img src="./165c8f261e76ac9a6c271a1ca771db94bf73b274.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.96ex; height:2.843ex;" alt="{\displaystyle \rho :T\rightarrow GL(1;\mathbb {C} )=\mathbb {C} ^{*}.}" loading="lazy"></dd></dl>

Since, also, T is compact, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> must actually map into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}\subset \mathbb {C} }">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}\subset \mathbb {C} }</annotation>

</semantics>

</math><img src="./ae6da5678e16d70116db48d2c1c312ac626bbc5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.352ex; height:2.676ex;" alt="{\displaystyle S^{1}\subset \mathbb {C} }" loading="lazy">.

To describe these representations concretely, we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy"> be the Lie algebra of T and we write points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h\in T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h\in T}</annotation>

</semantics>

</math><img src="./b6f99ed631d704c476f8d7691c0be7f98dd4cf1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.816ex; height:2.176ex;" alt="{\displaystyle h\in T}" loading="lazy"> as

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h=e^{H},\quad H\in {\mathfrak {t}}.}</annotation>

</semantics>

</math><img src="./4289f5b11a60f2c36acf54dafaf94aed924dac5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.929ex; height:3.009ex;" alt="{\displaystyle h=e^{H},\quad H\in {\mathfrak {t}}.}" loading="lazy"></dd></dl>

In such coordinates, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> will have the form

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (e^{H})=e^{i\lambda (H)}}</annotation>

</semantics>

</math><img src="./e49aa39c5220203a1690ae914f22ad74adc554c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.465ex; height:3.343ex;" alt="{\displaystyle \rho (e^{H})=e^{i\lambda (H)}}" loading="lazy"></dd></dl>

for some linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy">.

Now, since the exponential map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\mapsto e^{H}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\mapsto e^{H}}</annotation>

</semantics>

</math><img src="./2d51d6270adaa013dce90d9b66d5e873bb9f817d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.453ex; height:2.676ex;" alt="{\displaystyle H\mapsto e^{H}}" loading="lazy"> is not injective, not every such linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> gives rise to a well-defined map of T into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>

</semantics>

</math><img src="./60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy">. Rather, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>

</semantics>

</math><img src="./4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> denote the kernel of the exponential map:

<mrow>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Gamma =\left\{H\in {\mathfrak {t}}\mid e^{2\pi H}=\operatorname {Id} \right\},}</annotation>

</semantics>

</math><img src="./17f16dc5863cec81f546da1d533f00e3e6b45c20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.716ex; height:3.343ex;" alt="{\displaystyle \Gamma =\left\{H\in {\mathfrak {t}}\mid e^{2\pi H}=\operatorname {Id} \right\},}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Id} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Id} }</annotation>

</semantics>

</math><img src="./87a028dce1d3343c8a9e04d20660c1abebae8737.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle \operatorname {Id} }" loading="lazy"> is the <a href="Identity_element" title="Identity element">identity element</a> of T. (We scale the exponential map here by a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"> in order to avoid such factors elsewhere.)

Then for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> to give a well-defined map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> must satisfy

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma ,}</annotation>

</semantics>

</math><img src="./a69f6326f4badc022bf43efc1e2c512b48416079.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.98ex; height:2.843ex;" alt="{\displaystyle \lambda (H)\in \mathbb {Z} ,\quad H\in \Gamma ,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

</semantics>

</math><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"> is the set of integers.<a href="#cite_note-9">[9]</a> A linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> satisfying this condition is called an analytically integral element. This integrality condition is related to, but not identical to, the notion of <a href="Weight_(representation_theory)#Weights_in_the_representation_theory_of_semisimple_Lie_algebras" title="Weight (representation theory)">integral element</a> in the setting of semisimple Lie algebras.<a href="#cite_note-10">[10]</a>

Suppose, for example, T is just the group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S^{1}}</annotation>

</semantics>

</math><img src="./60796c8d0c03cf575637d3202463b214d9635880.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1}}" loading="lazy"> of complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\theta }}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e^{i\theta }}</annotation>

</semantics>

</math><img src="./7b4b8e67ee479d68e0e5040aaf87eff99214c90f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.654ex; height:2.676ex;" alt="{\displaystyle e^{i\theta }}" loading="lazy"> of absolute value 1. The Lie algebra is the set of purely imaginary numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,}</annotation>

</semantics>

</math><img src="./2daeaa9ea3754be35f3384761302e35b1105cdb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.732ex; height:2.509ex;" alt="{\displaystyle H=i\theta ,\,\theta \in \mathbb {R} ,}" loading="lazy"> and the kernel of the (scaled) exponential map is the set of numbers of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle in}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle in}</annotation>

</semantics>

</math><img src="./bbb5408e1e5640483e849a62c2cbdff6ae29cff5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle in}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> is an integer. A linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> takes integer values on all such numbers if and only if it is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (i\theta )=k\theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (i\theta )=k\theta }</annotation>

</semantics>

</math><img src="./d06ce9f62b94033f6f1e427b54bae1e78ac6f949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.458ex; height:2.843ex;" alt="{\displaystyle \lambda (i\theta )=k\theta }" loading="lazy"> for some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. The irreducible representations of T in this case are one-dimensional and of the form

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho (e^{i\theta })=e^{ik\theta },\quad k\in \mathbb {Z} .}</annotation>

</semantics>

</math><img src="./a403064614db59526e7bcb116128a713c08ce469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.88ex; height:3.176ex;" alt="{\displaystyle \rho (e^{i\theta })=e^{ik\theta },\quad k\in \mathbb {Z} .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Representation_theory_of_K">Representation theory of K</h3></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras" title="Lie algebra representation">Lie algebra representation § Classifying finite-dimensional representations of Lie algebras</a></div>

We now let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> denote a finite-dimensional irreducible representation of K (over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation>

</semantics>

</math><img src="./f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy">). We then consider the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> to T. This restriction is not irreducible unless <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> is one-dimensional. Nevertheless, the restriction decomposes as a direct sum of irreducible representations of T. (Note that a given irreducible representation of T may occur more than once.) Now, each irreducible representation of T is described by a linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> as in the preceding subsection. If a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> occurs at least once in the decomposition of the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> to T, we call <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> a weight of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy">. The strategy of the representation theory of K is to classify the irreducible representations in terms of their weights.

We now briefly describe the structures needed to formulate the theorem; more details can be found in the article on <a href="Weight_(representation_theory)#Weights_in_the_representation_theory_of_semisimple_Lie_algebras" title="Weight (representation theory)">weights in representation theory</a>. We need the notion of a root system for K (relative to a given maximal torus T). The construction of this root system <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subset {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\subset {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./0bae227fa27b854288e5a1b5d43e53319db6a813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle R\subset {\mathfrak {t}}}" loading="lazy"> is very similar to the <a href="Semisimple_Lie_algebra#Cartan_subalgebras_and_root_systems" title="Semisimple Lie algebra">construction for complex semisimple Lie algebras</a>. Specifically, the weights are the nonzero weights for the adjoint action of T on the complexified Lie algebra of K. The root system R has all the usual properties of a <a href="Root_system" title="Root system">root system</a>, except that the elements of R may not span <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {t}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {t}}}</annotation>

</semantics>

</math><img src="./7aabd0d28bc7f5b43a8b64d2a529308ca100e8ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.809ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {t}}}" loading="lazy">.<a href="#cite_note-11">[11]</a> We then choose a base <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> for R and we say that an integral element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is dominant if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda (\alpha )\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda (\alpha )\geq 0}</annotation>

</semantics>

</math><img src="./b01c687b258afe7865cb1bf0a78a7dcb5b0628b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.913ex; height:2.843ex;" alt="{\displaystyle \lambda (\alpha )\geq 0}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \in \Delta }</annotation>

</semantics>

</math><img src="./884bf3ccfc00b9bdc7941247a399a01fa200dc1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.264ex; height:2.176ex;" alt="{\displaystyle \alpha \in \Delta }" loading="lazy">. Finally, we say that one weight is higher than another if their difference can be expressed as a linear combination of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> with non-negative coefficients.

The irreducible finite-dimensional representations of K are then classified by a theorem of the highest <a href="Weight_(representation_theory)" title="Weight (representation theory)">weight</a>,<a href="#cite_note-12">[12]</a> which is closely related to the analogous theorem classifying <a href="Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras" title="Lie algebra representation">representations of a semisimple Lie algebra</a>. The result says that:

<ol><li>every irreducible representation has highest weight,</li>

<li>the highest weight is always a dominant, analytically integral element,</li>

<li>two irreducible representations with the same highest weight are isomorphic, and</li>

<li>every dominant, analytically integral element arises as the highest weight of an irreducible representation.</li></ol>

The theorem of the highest weight for representations of K is then almost the same as for semisimple Lie algebras, with one notable exception: The concept of an <a href="Weight_(representation_theory)#Integral_weight" title="Weight (representation theory)">integral element</a> is different. The weights <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> of a representation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Sigma }</annotation>

</semantics>

</math><img src="./9e1f558f53cda207614abdf90162266c70bc5c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Sigma }" loading="lazy"> are analytically integral in the sense described in the previous subsection. Every analytically integral element is <a href="Weight_(representation_theory)#Integral_weight" title="Weight (representation theory)">integral</a> in the Lie algebra sense, but not the other way around.<a href="#cite_note-13">[13]</a> (This phenomenon reflects that, in general, <a href="Lie_group%E2%80%93Lie_algebra_correspondence#Lie_group_representations" title="Lie group–Lie algebra correspondence">not every representation</a> of the Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {k}}}</annotation>

</semantics>

</math><img src="./a94eb54c7bdae2f76ad4d43f210dd71b5fa2beb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {k}}}" loading="lazy"> comes from a representation of the group K.) On the other hand, if K is simply connected, the set of possible highest weights in the group sense is the same as the set of possible highest weights in the Lie algebra sense.<a href="#cite_note-14">[14]</a>

<div class="mw-heading mw-heading3"><h3 id="The_Weyl_character_formula">The Weyl character formula</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi :K\to \operatorname {GL} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi :K\to \operatorname {GL} (V)}</annotation>

</semantics>

</math><img src="./b4fbdc395d627aca4fc924a8f899c53aaea9c5fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.234ex; height:2.843ex;" alt="{\displaystyle \Pi :K\to \operatorname {GL} (V)}" loading="lazy"> is representation of K, we define the character of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> to be the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} :K\to \mathbb {C} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} :K\to \mathbb {C} }</annotation>

</semantics>

</math><img src="./f1981bbe4f0a24e87b6411cce81b19e008efcbbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.038ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} :K\to \mathbb {C} }" loading="lazy"> given by

</mrow>

<mi>trace</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} (x)=\operatorname {trace} (\Pi (x)),\quad x\in K}</annotation>

</semantics>

</math><img src="./8ad38457511d179c8fc79ee3b7eb77eedd64011f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.308ex; height:2.843ex;" alt="{\displaystyle \mathrm {X} (x)=\operatorname {trace} (\Pi (x)),\quad x\in K}" loading="lazy">.</dd></dl>

This function is easily seen to be a class function, i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)}">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)}</annotation>

</semantics>

</math><img src="./c8d6e5ba204e3a08463aca35a75ecabba63bac2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.506ex; height:3.176ex;" alt="{\displaystyle \mathrm {X} (xyx^{-1})=\mathrm {X} (y)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> in K. Thus, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} }</annotation>

</semantics>

</math><img src="./a8d98cdc2216c7394d189ea3e09a479c826263b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} }" loading="lazy"> is determined by its restriction to T.

The study of characters is an important part of the representation theory of compact groups. One crucial result, which is a corollary of the <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a>, is that the characters form an orthonormal basis for the set of square-integrable class functions in K. A second key result is the <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a>, which gives an explicit formula for the character—or, rather, the restriction of the character to T—in terms of the highest weight of the representation.

In the closely related representation theory of semisimple Lie algebras, the Weyl character formula is an additional result established after the representations have been classified. In Weyl's analysis of the compact group case, however, the Weyl character formula is actually a crucial part of the classification itself. Specifically, in Weyl's analysis of the representations of K, the hardest part of the theorem—showing that every dominant, analytically integral element is actually the highest weight of some representation—is proved in a totally different way from the usual Lie algebra construction using <a href="Verma_module" title="Verma module">Verma modules</a>. In Weyl's approach, the construction is based on the <a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a> and an analytic proof of the <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a>.<a href="#cite_note-15">[15]</a> Ultimately, the irreducible representations of K are realized inside the space of continuous functions on K.

We now consider the case of the compact group SU(2). The representations are often considered from the <a href="Representation_theory_of_SU(2)" title="Representation theory of SU(2)">Lie algebra point of view</a>, but we here look at them from the group point of view. We take the maximal torus to be the set of matrices of the form

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}.}</annotation>

</semantics>

</math><img src="./a82d3c9fb33988ac997396ca51d9a43a5124a5fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.729ex; height:6.509ex;" alt="{\displaystyle {\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}.}" loading="lazy"></dd></dl>

According to the example discussed above in the section on representations of T, the analytically integral elements are labeled by integers, so that the dominant, analytically integral elements are non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">. The general theory then tells us that for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">, there is a unique irreducible representation of SU(2) with highest weight <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">.

Much information about the representation corresponding to a given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> is encoded in its character. Now, the Weyl character formula says, <a href="Weyl_character_formula#The_SU.282.29_case" title="Weyl character formula">in this case</a>, that the character is given by

</mrow>

<mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\sin((m+1)\theta )}{\sin(\theta )}}.}</annotation>

</semantics>

</math><img src="./04c365b1bd41f9cda2aaa18234709e9d8c379ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:36.823ex; height:6.509ex;" alt="{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)={\frac {\sin((m+1)\theta )}{\sin(\theta )}}.}" loading="lazy"></dd></dl>

We can also write the character as sum of exponentials as follows:

</mrow>

<mrow>

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta }+e^{-im\theta }.}</annotation>

</semantics>

</math><img src="./309c0181e21b0e68ec8c5f67ac0680e90b6b6fa5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.715ex; height:6.509ex;" alt="{\displaystyle \mathrm {X} \left({\begin{pmatrix}e^{i\theta }&0\\0&e^{-i\theta }\end{pmatrix}}\right)=e^{im\theta }+e^{i(m-2)\theta }+\cdots e^{-i(m-2)\theta }+e^{-im\theta }.}" loading="lazy"></dd></dl>

(If we use the formula for the sum of a finite geometric series on the above expression and simplify, we obtain the earlier expression.)

From this last expression and the standard formula for the <a href="Weyl_character_formula#Complex_semisimple_Lie_algebras" title="Weyl character formula">character in terms of the weights of the representation</a>, we can read off that the weights of the representation are

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,m-2,\ldots ,-(m-2),-m,}</annotation>

</semantics>

</math><img src="./e0a28bf95895ecab63b391cbaaa4dbaebb412238.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.486ex; height:2.843ex;" alt="{\displaystyle m,m-2,\ldots ,-(m-2),-m,}" loading="lazy"></dd></dl>

each with multiplicity one. (The weights are the integers appearing in the exponents of the exponentials and the multiplicities are the coefficients of the exponentials.) Since there are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m+1}</annotation>

</semantics>

</math><img src="./c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}" loading="lazy"> weights, each with multiplicity 1, the dimension of the representation is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m+1}</annotation>

</semantics>

</math><img src="./c6f7ed29a2b4a62d3b6af05cd91a58ffc6094201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.043ex; height:2.343ex;" alt="{\displaystyle m+1}" loading="lazy">. Thus, we recover much of the information about the representations that is usually obtained from the Lie algebra computation.

<div class="mw-heading mw-heading3"><h3 id="An_outline_of_the_proof">An outline of the proof</h3></div>

We now outline the proof of the theorem of the highest weight, following the original argument of <a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a>. We continue to let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> be a connected compact Lie group and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> a fixed maximal torus in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. We focus on the most difficult part of the theorem, showing that every dominant, analytically integral element is the highest weight of some (finite-dimensional) irreducible representation.<a href="#cite_note-16">[16]</a>

The tools for the proof are the following:

<ul><li>The <a href="Maximal_torus#Properties" title="Maximal torus">torus theorem</a>.</li>

<li>The <a href="Maximal_torus#Weyl_integral_formula" title="Maximal torus">Weyl integral formula</a>.</li>

<li>The <a href="Peter%E2%80%93Weyl_theorem#Restriction_to_class_functions" title="Peter–Weyl theorem">Peter–Weyl theorem for class functions</a>, which states that the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">.</li></ul>

With these tools in hand, we proceed with the proof. The first major step in the argument is to prove the <a href="Weyl_character_formula" title="Weyl character formula">Weyl character formula</a>. The formula states that if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> is an irreducible representation with highest weight <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy">, then the character <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {X} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} }</annotation>

</semantics>

</math><img src="./a8d98cdc2216c7394d189ea3e09a479c826263b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \mathrm {X} }" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Pi }</annotation>

</semantics>

</math><img src="./eed3e3db6cc2028a183af948212ed2551d25c954.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle \Pi }" loading="lazy"> satisfies:

</mrow>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {X} (e^{H})={\frac {\sum _{w\in W}\det(w)e^{i\langle w\cdot (\lambda +\rho ),H\rangle }}{\sum _{w\in W}\det(w)e^{i\langle w\cdot \rho ,H\rangle }}}}</annotation>

</semantics>

</math><img src="./b7a38fac46d48ccc045b20ca14cc7b9857c8c576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:35.112ex; height:7.343ex;" alt="{\displaystyle \mathrm {X} (e^{H})={\frac {\sum _{w\in W}\det(w)e^{i\langle w\cdot (\lambda +\rho ),H\rangle }}{\sum _{w\in W}\det(w)e^{i\langle w\cdot \rho ,H\rangle }}}}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"> in the Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is half the sum of the positive roots. (The notation uses the convention of "real weights"; this convention requires an explicit factor of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"> in the exponent.) Weyl's proof of the character formula is analytic in nature and hinges on the fact that the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{2}}</annotation>

</semantics>

</math><img src="./ba162c66ca85776c83557af5088cc6f8584d1912.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{2}}" loading="lazy"> norm of the character is 1. Specifically, if there were any additional terms in the numerator, the Weyl integral formula would force the norm of the character to be greater than 1.

Next, we let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy"> denote the function on the right-hand side of the character formula. We show that even if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is not known to be the highest weight of a representation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy"> is a well-defined, Weyl-invariant function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy">, which therefore extends to a class function on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">. Then using the Weyl integral formula, one can show that as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> ranges over the set of dominant, analytically integral elements, the functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy"> form an orthonormal family of class functions. We emphasize that we do not currently know that every such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is the highest weight of a representation; nevertheless, the expressions on the right-hand side of the character formula gives a well-defined set of functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy">, and these functions are orthonormal.

Now comes the conclusion. The set of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy">—with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> ranging over the dominant, analytically integral elements—forms an orthonormal set in the space of square integrable class functions. But by the Weyl character formula, the characters of the irreducible representations form a subset of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy">'s. And by the Peter–Weyl theorem, the characters of the irreducible representations form an orthonormal basis for the space of square integrable class functions. If there were some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> that is not the highest weight of a representation, then the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy"> would not be the character of a representation. Thus, the characters would be a proper subset of the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy">'s. But then we have an impossible situation: an orthonormal basis (the set of characters of the irreducible representations) would be contained in a strictly larger orthonormal set (the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\lambda }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi _{\lambda }}</annotation>

</semantics>

</math><img src="./6d79124b083c0354b2914bd3562be6e5d2caff28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.869ex; height:2.509ex;" alt="{\displaystyle \Phi _{\lambda }}" loading="lazy">'s). Thus, every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> must actually be the highest weight of a representation.

<div class="mw-heading mw-heading2"><h2 id="Duality">Duality</h2></div>

The topic of recovering a compact group from its representation theory is the subject of the <a href="Tannaka%E2%80%93Krein_duality" title="Tannaka–Krein duality">Tannaka–Krein duality</a>, now often recast in terms of <a href="Tannakian_category" class="mw-redirect" title="Tannakian category">Tannakian category</a> theory.

<div class="mw-heading mw-heading2"><h2 id="From_compact_to_non-compact_groups">From compact to non-compact groups</h2></div>

The influence of the compact group theory on non-compact groups was formulated by Weyl in his <a href="Unitarian_trick" title="Unitarian trick">unitarian trick</a>. Inside a general <a href="Semisimple_Lie_group" class="mw-redirect" title="Semisimple Lie group">semisimple Lie group</a> there is a <a href="Maximal_compact_subgroup" title="Maximal compact subgroup">maximal compact subgroup</a>, and the representation theory of such groups, developed largely by <a href="Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a>, uses intensively the <a href="Restriction_of_a_representation" class="mw-redirect" title="Restriction of a representation">restriction of a representation</a> to such a subgroup, and also the model of Weyl's character theory.

<ul><li><a href="Peter%E2%80%93Weyl_theorem" title="Peter–Weyl theorem">Peter–Weyl theorem</a></li>

<li><a href="Maximal_torus" title="Maximal torus">Maximal torus</a></li>

<li><a href="Root_system" title="Root system">Root system</a></li>

<li><a href="Locally_compact_group" title="Locally compact group">Locally compact group</a></li>

<li><a href="P-compact_group" title="P-compact group">p-compact group</a></li>

<li><a href="Protorus" title="Protorus">Protorus</a></li>

<li><a href="Lie_algebra_representation#Classifying_finite-dimensional_representations_of_Lie_algebras" title="Lie algebra representation">Classifying finite-dimensional representations of Lie algebras</a></li>

<li><a href="Weight_(representation_theory)#Weights_in_the_representation_theory_of_semisimple_Lie_algebras" title="Weight (representation theory)">Weights in the representation theory of semisimple Lie algebras</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 1.2

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFBröckertom_Dieck1985">Bröcker & tom Dieck 1985</a>, Chapter V, Sections 7 and 8

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Chapter 11

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 7.7

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 13.8

</li>

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</style><cite id="CITEREFWeil1940" class="citation cs2"><a href="Andr%C3%A9_Weil" title="André Weil">Weil, André</a> (1940), L'intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, vol. 869, Paris: Hermann</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFPeterWeyl1927" class="citation cs2">Peter, F.; Weyl, H. (1927), "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe", Math. Ann., 97: 737–755, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01447892">10.1007/BF01447892</a></cite>.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Part III

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Proposition 12.9

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 12.2

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 11.7

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Chapter 12

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Section 12.2

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Corollary 13.20

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Sections 12.4 and 12.5

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFHall2015">Hall 2015</a> Sections 12.4 and 12.5

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFBröckertom_Dieck1985" class="citation cs2">Bröcker, Theodor; tom Dieck, Tammo (1985), Representations of Compact Lie Groups, Graduate Texts in Mathematics, vol. 98, Springer</cite></li>

<li><cite id="CITEREFHall2015" class="citation cs2">Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3319134666</bdi></cite></li>

<li><cite id="CITEREFHofmannMorris1998" class="citation cs2">Hofmann, Karl H.; Morris, Sidney A. (1998), The structure of compact groups, Berlin: de Gruyter, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-11-015268-1</bdi></cite></li></ul></div><div><div class="zim-footer">

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