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border-bottom-width: 1px; font-size:95%; margin-bottom:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="Vorlage_Begriffsklärungshinweis"><div class="noviewer noresize bksicon" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" aria-hidden="true" role="presentation"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/25px-Disambig-dark.svg.png" decoding="async" width="25" height="19" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/38px-Disambig-dark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Disambig-dark.svg/50px-Disambig-dark.svg.png 2x" data-file-width="444" data-file-height="340" /></span></span></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div role="navigation"> Der Titel dieses Artikels ist mehrdeutig. Weitere Bedeutungen sind unter <a href="/wiki/Darstellungstheorie_(Begriffskl%C3%A4rung)" class="mw-disambig" title="Darstellungstheorie (Begriffsklärung)">Darstellungstheorie (Begriffsklärung)</a> aufgeführt.</div> </div></div> <p>In der <b>Darstellungstheorie</b> werden Elemente von <a href="/wiki/Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a> oder allgemeiner von <a href="/wiki/Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebren</a> mittels <a href="/wiki/Homomorphismus" title="Homomorphismus">Homomorphismen</a> auf <a href="/wiki/Lineare_Abbildung" title="Lineare Abbildung">lineare Abbildungen</a> von Vektorräumen (<a href="/wiki/Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrizen</a>) abgebildet. </p><p>Die Darstellungstheorie hat Anwendungen in fast allen Gebieten der <a href="/wiki/Mathematik" title="Mathematik">Mathematik</a> und der <a href="/wiki/Theoretische_Physik" class="mw-redirect" title="Theoretische Physik">theoretischen Physik</a>. So war ein darstellungstheoretischer Satz von <a href="/wiki/Robert_Langlands" title="Robert Langlands">Robert Langlands</a> ein wesentlicher Schritt für <a href="/wiki/Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>’ Beweis des <a href="/wiki/Gro%C3%9Fer_Satz_von_Fermat" class="mw-redirect" title="Großer Satz von Fermat">Großen Satzes von Fermat</a>, und die Darstellungstheorie lieferte ebenfalls den theoretischen Hintergrund für die Vorhersage, dass <a href="/wiki/Quark_(Physik)" title="Quark (Physik)">Quarks</a> existieren.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Auch für die rein algebraische Untersuchung der Gruppen oder Algebren ist die Darstellung durch Matrizen oft nützlich. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Arten_von_Darstellungen"><span class="tocnumber">1</span> <span class="toctext">Arten von Darstellungen</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Grundbegriffe"><span class="tocnumber">2</span> <span class="toctext">Grundbegriffe</span></a> <ul> <li class="toclevel-2 tocsection-3"><a href="#Teildarstellungen"><span class="tocnumber">2.1</span> <span class="toctext">Teildarstellungen</span></a></li> <li class="toclevel-2 tocsection-4"><a href="#Direkte_Summen"><span class="tocnumber">2.2</span> <span class="toctext">Direkte Summen</span></a></li> <li class="toclevel-2 tocsection-5"><a href="#Irreduzibilität,_Vollständige_Reduzibilität,_Ausreduzierung"><span class="tocnumber">2.3</span> <span class="toctext">Irreduzibilität, Vollständige Reduzibilität, Ausreduzierung</span></a></li> </ul> </li> <li class="toclevel-1 tocsection-6"><a href="#Geschichte"><span class="tocnumber">3</span> <span class="toctext">Geschichte</span></a></li> <li class="toclevel-1 tocsection-7"><a href="#Literatur"><span class="tocnumber">4</span> <span class="toctext">Literatur</span></a></li> <li class="toclevel-1 tocsection-8"><a href="#Weblinks"><span class="tocnumber">5</span> <span class="toctext">Weblinks</span></a></li> <li class="toclevel-1 tocsection-9"><a href="#Einzelnachweise"><span class="tocnumber">6</span> <span class="toctext">Einzelnachweise</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Arten_von_Darstellungen">Arten von Darstellungen</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=1" title="Abschnitt bearbeiten: Arten von Darstellungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Arten von Darstellungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Klassisch beschäftigte sich die Darstellungstheorie mit Homomorphismen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon G\rightarrow GL(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \colon G\rightarrow GL(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0376ac3b51602ae184367ee20898fd4c82265eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.683ex; height:2.843ex;" alt="{\displaystyle \rho \colon G\rightarrow GL(V)}"></span> für Gruppen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" 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}"></span> und <a href="/wiki/Vektorraum" title="Vektorraum">Vektorräume</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle GL(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle GL(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ad853a4ebab014e00fbd05a7a75beca186ed9a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.006ex; height:2.843ex;" alt="{\displaystyle GL(V)}"></span> die <a href="/wiki/Allgemeine_lineare_Gruppe" title="Allgemeine lineare Gruppe">allgemeine lineare Gruppe</a> über <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> bezeichnet), siehe </p> <ul><li><a href="/wiki/Darstellung_(Gruppe)" title="Darstellung (Gruppe)">Darstellung (Gruppe)</a>.</li></ul> <p>Allgemeiner wird die Darstellungstheorie von Ringen und Algebren betrachtet, welche die Darstellungstheorie der Gruppen als Spezialfall enthält (weil jede Darstellung einer Gruppe eine Darstellung ihres <a href="/wiki/Gruppenring" class="mw-redirect" title="Gruppenring">Gruppenringes</a> induziert), hierfür siehe </p> <ul><li><a href="/wiki/Darstellung_(Algebra)" title="Darstellung (Algebra)">Darstellung (Algebra)</a>.</li></ul> <p>In der Physik sind neben den diskreten Gruppen der <a href="/wiki/Festk%C3%B6rperphysik" title="Festkörperphysik">Festkörperphysik</a> besonders auch Darstellungen von <a href="/wiki/Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppen</a> von Bedeutung, etwa bei der <a href="/wiki/Drehgruppe" title="Drehgruppe">Drehgruppe</a> und den Gruppen des <a href="/wiki/Standardmodell" title="Standardmodell">Standardmodells</a>. Hier verlangt man zusätzlich, dass Darstellungen <a href="/wiki/Differenzierbarkeit" title="Differenzierbarkeit">glatte</a> Homomorphismen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon G\rightarrow \operatorname {GL} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>:</mo> <mi>G</mi> <mo stretchy="false">→</mo> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \colon G\rightarrow \operatorname {GL} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94063f412fb466db38371e6b7f03e63ad73a840d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.55ex; height:2.843ex;" alt="{\displaystyle \rho \colon G\rightarrow \operatorname {GL} (V)}"></span> sein sollen, siehe </p> <ul><li><a href="/wiki/Darstellung_(Lie-Gruppe)" class="mw-redirect" title="Darstellung (Lie-Gruppe)">Darstellung (Lie-Gruppe)</a>.</li></ul> <p>Die <a href="/wiki/Liesche_S%C3%A4tze" title="Liesche Sätze">Lieschen Sätze</a> vermitteln eine Korrespondenz zwischen Darstellungen von Lie-Gruppen und den induzierten Darstellungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi =D_{e}\rho \colon {\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mi>ρ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi =D_{e}\rho \colon {\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db19db30c8ceee76bbbb4a09ea40408dc136b7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.795ex; height:2.843ex;" alt="{\displaystyle \pi =D_{e}\rho \colon {\mathfrak {g}}\rightarrow {\mathfrak {gl}}(V)}"></span> <a href="/wiki/Lie-Gruppe#Lie-Algebra_der_Lie-Gruppe" title="Lie-Gruppe">ihrer Lie-Algebren</a>. Für die Darstellungstheorie von <a href="/wiki/Lie-Algebren" class="mw-redirect" title="Lie-Algebren">Lie-Algebren</a> siehe </p> <ul><li><a href="/wiki/Darstellung_(Lie-Algebra)" title="Darstellung (Lie-Algebra)">Darstellung (Lie-Algebra)</a>.</li></ul> <p>Lie-Algebren sind nicht assoziativ, weshalb ihre Darstellungstheorie kein Spezialfall der Darstellungstheorie assoziativer Algebren ist. Man kann aber jeder Lie-Algebra ihre <a href="/wiki/Universelle_einh%C3%BCllende_Algebra" title="Universelle einhüllende Algebra">universelle einhüllende Algebra</a> zuordnen, welche eine assoziative Algebra ist. </p><p>Bei <a href="/wiki/Banach-*-Algebra" class="mw-redirect" title="Banach-*-Algebra">Banach-*-Algebren</a> wie <a href="/wiki/C*-Algebra" title="C*-Algebra">C<sup>*</sup>-Algebren</a> oder <a href="/wiki/Gruppen-C*-Algebra#Die_Gruppenalgebra" title="Gruppen-C*-Algebra">Gruppenalgebren</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10e36f941f2b8fec8f9c058ebefd8ac69609b97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.273ex; height:3.176ex;" alt="{\displaystyle L^{1}(G)}"></span> verwendet man als Vektorräume für Darstellungen dieser Algebren in natürlicher Weise <a href="/wiki/Hilbertraum" title="Hilbertraum">Hilberträume</a>, siehe </p> <ul><li><a href="/wiki/Hilbertraum-Darstellung" title="Hilbertraum-Darstellung">Hilbertraum-Darstellung</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Grundbegriffe">Grundbegriffe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=2" title="Abschnitt bearbeiten: Grundbegriffe" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Grundbegriffe"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Im Folgenden sei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span> eine Gruppe, Lie-Gruppe oder Algebra und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon A\rightarrow L(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \colon A\rightarrow L(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b5929a99ccd844c8d526d7391f21ee7be9c67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.902ex; height:2.843ex;" alt="{\displaystyle \pi \colon A\rightarrow L(V)}"></span> eine Darstellung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span>, also ein Gruppen-, Lie-Gruppen- oder Algebren-Homomorphismus in die Algebra <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f042ad62c43afc18dabfb243307f9b8ccaf94cff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle L(V)}"></span> der linearen Abbildungen eines Vektorraums <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> (dessen Bild im Falle von Gruppen- oder Lie-Gruppen-Isomorphismen natürlich sogar in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} (V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>GL</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} (V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea57e2326666ffe7efa076e1e93e01aab303d556" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.874ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} (V)}"></span> liegt). </p><p>Die <a href="/wiki/Dimension_(Mathematik)" title="Dimension (Mathematik)">Vektorraumdimension</a> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> wird als <i>Dimension</i> von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> bezeichnet. Endlichdimensionale Darstellungen nennt man auch <i>Matrix-Darstellungen</i>, denn durch Wahl einer <a href="/wiki/Vektorraumbasis" class="mw-redirect" title="Vektorraumbasis">Vektorraumbasis</a> lässt sich jedes Element aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f042ad62c43afc18dabfb243307f9b8ccaf94cff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.179ex; height:2.843ex;" alt="{\displaystyle L(V)}"></span> als Matrix schreiben. Injektive Darstellungen heißen <i>treu</i>. </p><p>Zwei Darstellungen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1c74f716bfe7d7b1bf7bd5581e716b14404531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.572ex; height:2.843ex;" alt="{\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bbf4f4a4a1856ae134428da2c9fdcf282135c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.572ex; height:2.843ex;" alt="{\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}"></span> heißen <i>äquivalent</i>, wenn es einen <a href="/wiki/Lineare_Abbildung" title="Lineare Abbildung">Vektorraum-Isomorphismus</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\colon V_{1}\rightarrow V_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\colon V_{1}\rightarrow V_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55a0db0f03c2176c964f7fb18b47014a2026c33e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.103ex; height:2.509ex;" alt="{\displaystyle T\colon V_{1}\rightarrow V_{2}}"></span> gibt mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(a)=T^{-1}\circ \pi _{2}(a)\circ T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>∘</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(a)=T^{-1}\circ \pi _{2}(a)\circ T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e045cdbc65b7d125a7b40c9c29e0eaf97edfee0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.014ex; height:3.176ex;" alt="{\displaystyle \pi _{1}(a)=T^{-1}\circ \pi _{2}(a)\circ T}"></span> für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span>. Dafür schreibt man abkürzend auch <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}\sim \pi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}\sim \pi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f804e1a0b73b7c35de3a022b85d21b872925b796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.857ex; height:2.009ex;" alt="{\displaystyle \pi _{1}\sim \pi _{2}}"></span>. Die so definierte Äquivalenz ist eine <a href="/wiki/%C3%84quivalenzrelation" title="Äquivalenzrelation">Äquivalenzrelation</a> auf der Klasse aller Darstellungen. Die Begriffsbildungen in der Darstellungstheorie sind so angelegt, dass sie beim Übergang zu einer äquivalenten Darstellung erhalten bleiben, Dimension und Treue sind erste Beispiele. </p> <div class="mw-heading mw-heading3"><h3 id="Teildarstellungen">Teildarstellungen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=3" title="Abschnitt bearbeiten: Teildarstellungen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Teildarstellungen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon A\rightarrow L(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \colon A\rightarrow L(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b5929a99ccd844c8d526d7391f21ee7be9c67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.902ex; height:2.843ex;" alt="{\displaystyle \pi \colon A\rightarrow L(V)}"></span> eine Darstellung. Ein <a href="/wiki/Untervektorraum" title="Untervektorraum">Untervektorraum</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\subset V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⊂</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\subset V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c7e1261dc1fa6e6a0315844eacf56653568e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle W\subset V}"></span> heißt invariant (genauer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span>-invariant), falls <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi (a)W\subset W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mi>W</mi> <mo>⊂</mo> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi (a)W\subset W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/708859ff79cd70cfc728af8e30a69d5bcc58c16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.34ex; height:2.843ex;" alt="{\displaystyle \pi (a)W\subset W}"></span> für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span>. </p><p>Offenbar ist </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\pi }}\colon A\rightarrow L(W),\quad a\mapsto {\tilde {\pi }}(a):=\pi (a)|_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>π</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>W</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>π</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {\pi }}\colon A\rightarrow L(W),\quad a\mapsto {\tilde {\pi }}(a):=\pi (a)|_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3383e95ee77ec40ae0634998a2779cc0f54daa3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.84ex; height:3.009ex;" alt="{\displaystyle {\tilde {\pi }}\colon A\rightarrow L(W),\quad a\mapsto {\tilde {\pi }}(a):=\pi (a)|_{W}}"></span></dd></dl> <p>wieder eine Darstellung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span>, die man die Einschränkung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }"></span> auf <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> nennt und mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi |_{W}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi |_{W}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306b0329ed44a638d32bcd47ea26aa68e66e1873" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.933ex; height:3.009ex;" alt="{\displaystyle \pi |_{W}}"></span> bezeichnet. </p><p>Ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W^{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd214b16c61c67ae44d7589532f6813b7df769b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.452ex; height:2.343ex;" alt="{\displaystyle W^{c}}"></span> ein zu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> <a href="/wiki/Komplement%C3%A4rraum" title="Komplementärraum">komplementärer Unterraum</a>, der ebenfalls invariant ist, so gilt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \sim \pi |_{W}\oplus \pi |_{W^{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>∼</mo> <mi>π</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>W</mi> </mrow> </msub> <mo>⊕</mo> <mi>π</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \sim \pi |_{W}\oplus \pi |_{W^{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f691a94d0eacdcbea61fb0342b5a6e74f56ebc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.931ex; height:3.009ex;" alt="{\displaystyle \pi \sim \pi |_{W}\oplus \pi |_{W^{c}}}"></span>, wobei die Äquivalenz durch den Isomorphismus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\oplus W^{c}\rightarrow V,(w_{1},w_{2})\mapsto w_{1}+w_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo>⊕</mo> <msup> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>V</mi> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W\oplus W^{c}\rightarrow V,(w_{1},w_{2})\mapsto w_{1}+w_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87d959855157c866a4898ba4294efbd2110de6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.334ex; height:2.843ex;" alt="{\displaystyle W\oplus W^{c}\rightarrow V,(w_{1},w_{2})\mapsto w_{1}+w_{2}}"></span> vermittelt wird. </p> <div class="mw-heading mw-heading3"><h3 id="Direkte_Summen">Direkte Summen</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=4" title="Abschnitt bearbeiten: Direkte Summen" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Direkte Summen"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sind <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d1c74f716bfe7d7b1bf7bd5581e716b14404531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.572ex; height:2.843ex;" alt="{\displaystyle \pi _{1}\colon A\rightarrow L(V_{1})}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bbf4f4a4a1856ae134428da2c9fdcf282135c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.572ex; height:2.843ex;" alt="{\displaystyle \pi _{2}\colon A\rightarrow L(V_{2})}"></span> zwei Darstellungen, so definiert </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon A\rightarrow L(V_{1}\oplus V_{2}),\quad a\mapsto \pi _{1}(a)\oplus \pi _{2}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo stretchy="false">↦</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \colon A\rightarrow L(V_{1}\oplus V_{2}),\quad a\mapsto \pi _{1}(a)\oplus \pi _{2}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce31a6b0cd036ab82f15b4abacb58829a813264" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.652ex; height:2.843ex;" alt="{\displaystyle \pi \colon A\rightarrow L(V_{1}\oplus V_{2}),\quad a\mapsto \pi _{1}(a)\oplus \pi _{2}(a)}"></span></dd></dl> <p>wieder eine Darstellung von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" 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 A}"></span>, wobei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}(a)\oplus \pi _{2}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}(a)\oplus \pi _{2}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4c564ff465737a4a051487ddacf10bafa25e5df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.677ex; height:2.843ex;" alt="{\displaystyle \pi _{1}(a)\oplus \pi _{2}(a)}"></span> komponentenweise auf der <a href="/wiki/Direkte_Summe" title="Direkte Summe">direkten Summe</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{1}\oplus V_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{1}\oplus V_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76703d11b2d3f04f98733946388930f14c56769b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.659ex; height:2.509ex;" alt="{\displaystyle V_{1}\oplus V_{2}}"></span> operiert, das heißt <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi _{1}(a)\oplus \pi _{2}(a))(\xi _{1}\oplus \xi _{2}):=\pi _{1}(a)\xi _{1}\oplus \pi _{2}(a)\xi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:=</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi _{1}(a)\oplus \pi _{2}(a))(\xi _{1}\oplus \xi _{2}):=\pi _{1}(a)\xi _{1}\oplus \pi _{2}(a)\xi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e5266aee7b761b2110b9063a063caf3a7f5b7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.849ex; height:2.843ex;" alt="{\displaystyle (\pi _{1}(a)\oplus \pi _{2}(a))(\xi _{1}\oplus \xi _{2}):=\pi _{1}(a)\xi _{1}\oplus \pi _{2}(a)\xi _{2}}"></span> für alle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{i}\in V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}\in V_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e11f0bb26edeeb81c142e2b0ddd9856139ed6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.814ex; height:2.509ex;" alt="{\displaystyle \xi _{i}\in V_{i}}"></span>. Diese Darstellung nennt man die direkte Summe aus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/542cbd3dacd0a061d666ed7fc4ed7ad15b47444b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{1}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b4d39c450ad33d7c407aec6fff9f225463ac1f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.379ex; height:2.009ex;" alt="{\displaystyle \pi _{2}}"></span> und bezeichnet sie mit <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{1}\oplus \pi _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{1}\oplus \pi _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d492cf194c701d4d81bf8a3ea1a9c9b530bb0a35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.599ex; height:2.343ex;" alt="{\displaystyle \pi _{1}\oplus \pi _{2}}"></span>. </p><p>Diese Konstruktion lässt sich für direkte Summen beliebig vieler Summanden verallgemeinern. Ist <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\pi _{i})_{i\in I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\pi _{i})_{i\in I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6555407fdac1e57906e79eb7da3152204a51324f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.658ex; height:2.843ex;" alt="{\displaystyle (\pi _{i})_{i\in I}}"></span> eine Familie von Darstellungen, so auch </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{i\in I}\pi _{i}\colon A\rightarrow L\left(\bigoplus _{i\in I}V_{i}\right),\,a\mapsto \bigoplus _{i\in I}\pi _{i}(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mrow> <mo>(</mo> <mrow> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mi>a</mi> <mo stretchy="false">↦</mo> <munder> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \bigoplus _{i\in I}\pi _{i}\colon A\rightarrow L\left(\bigoplus _{i\in I}V_{i}\right),\,a\mapsto \bigoplus _{i\in I}\pi _{i}(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fb3ade1a688fa56c4b54614c81426c8063c38a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.831ex; height:7.509ex;" alt="{\displaystyle \bigoplus _{i\in I}\pi _{i}\colon A\rightarrow L\left(\bigoplus _{i\in I}V_{i}\right),\,a\mapsto \bigoplus _{i\in I}\pi _{i}(a)}"></span>.</dd></dl> <div class="mw-heading mw-heading3"><h3 id="Irreduzibilität,_Vollständige_Reduzibilität,_Ausreduzierung"><span id="Irreduzibilit.C3.A4t.2C_Vollst.C3.A4ndige_Reduzibilit.C3.A4t.2C_Ausreduzierung"></span>Irreduzibilität, Vollständige Reduzibilität, Ausreduzierung</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=5" title="Abschnitt bearbeiten: Irreduzibilität, Vollständige Reduzibilität, Ausreduzierung" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=5" title="Quellcode des Abschnitts bearbeiten: Irreduzibilität, Vollständige Reduzibilität, Ausreduzierung"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eine Darstellung <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon A\rightarrow L(V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mo>:</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>L</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi \colon A\rightarrow L(V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00b5929a99ccd844c8d526d7391f21ee7be9c67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.902ex; height:2.843ex;" alt="{\displaystyle \pi \colon A\rightarrow L(V)}"></span> heißt irreduzibel, wenn es außer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}"></span> und <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> keine weiteren invarianten Unterräume von <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> gibt. Für eine äquivalente Charakterisierung siehe <a href="/wiki/Lemma_von_Schur" title="Lemma von Schur">Lemma von Schur</a>. Eine Darstellung heißt vollständig reduzibel, wenn sie zu einer direkten Summe irreduzibler Darstellungen äquivalent ist. Das „Produkt“ (besser: <a href="/wiki/Tensorprodukt#Tensorprodukt_von_Darstellungen" title="Tensorprodukt">Tensorprodukt</a>) zweier (irreduzibler) Darstellungen ist i.a. reduzibel und kann nach Bestandteilen der irreduziblen Darstellungen „ausreduziert“ werden, wobei spezielle Koeffizienten wie z. B. die <a href="/wiki/Clebsch-Gordan-Koeffizient" title="Clebsch-Gordan-Koeffizient">Clebsch-Gordan-Koeffizienten</a> der Drehimpulsphysik entstehen. Dies ist für die Anwendungen in der Physik ein besonders wichtiger Aspekt. </p> <div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=6" title="Abschnitt bearbeiten: Geschichte" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=6" title="Quellcode des Abschnitts bearbeiten: Geschichte"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Im 18. und 19. Jahrhundert kamen Darstellungstheorie und Harmonische Analysis (in Form der Zerlegung von Funktionen in <a href="/wiki/Multiplikativer_Charakter" class="mw-redirect" title="Multiplikativer Charakter">multiplikative Charaktere</a>) abelscher Gruppen wie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" 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 {R} }"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd6ed3ed42d3ae16ae67a7049e8bad73dfb00e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.885ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} /2\pi \mathbb {Z} }"></span> oder <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /n\mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2120ebbc85f91df66c6de5446367bf9fd620844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }"></span> beispielsweise im Zusammenhang mit <a href="/wiki/Euler-Produkt" title="Euler-Produkt">Euler-Produkten</a> oder <a href="/wiki/Fourier-Transformation" title="Fourier-Transformation">Fourier-Transformationen</a> vor. Dabei arbeitete man aber nicht mit den Darstellungen, sondern mit deren multiplikativen Charakteren. Frobenius definierte 1896 zuerst (ohne explizit auf Darstellungen Bezug zu nehmen) einen Begriff multiplikativer Charaktere auch für nichtabelsche Gruppen, Burnside und Schur entwickelten seine Definitionen dann neu auf der Basis von Matrix-Darstellungen und <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> gab schließlich im Wesentlichen die heutige Definition mittels linearer Abbildungen eines Vektorraumes, was später die in der Quantenmechanik benötigte Untersuchung unendlich-dimensionaler Darstellungen ermöglichte. </p><p>Um 1900 wurde die Darstellungstheorie der <a href="/wiki/Symmetrische_Gruppe" title="Symmetrische Gruppe">symmetrischen</a> und <a href="/wiki/Alternierende_Gruppe" title="Alternierende Gruppe">alternierenden Gruppen</a> von Frobenius und Young ausgearbeitet. 1913 bewies Cartan den <a href="/wiki/Satz_vom_h%C3%B6chsten_Gewicht" title="Satz vom höchsten Gewicht">Satz vom höchsten Gewicht</a>, der die irreduziblen Darstellungen komplexer halbeinfacher Lie-Algebren klassifiziert. Schur beobachtete 1924, dass man mittels <a href="/w/index.php?title=Invariante_Integration&action=edit&redlink=1" class="new" title="Invariante Integration (Seite nicht vorhanden)">invarianter Integration</a> die <a href="/wiki/Darstellungstheorie_endlicher_Gruppen" title="Darstellungstheorie endlicher Gruppen">Darstellungstheorie endlicher Gruppen</a> auf kompakte Gruppen ausdehnen kann, die Darstellungstheorie kompakter zusammenhängender Lie-Gruppen wurde dann von Weyl entwickelt. Die von Haar und von Neumann bewiesene Existenz und Eindeutigkeit des <a href="/wiki/Haar-Ma%C3%9F" class="mw-redirect" title="Haar-Maß">Haar-Maßes</a> erlaubte dann Anfang der 30er Jahre die Erweiterung dieser Theorie auf kompakte topologische Gruppen. Weitere Entwicklungen betrafen dann die Anwendung der Darstellungstheorie lokal kompakter Gruppen wie der <a href="/wiki/Heisenberggruppe" class="mw-redirect" title="Heisenberggruppe">Heisenberggruppe</a> in der Quantenmechanik, die Theorie lokal kompakter abelscher Gruppen mit Anwendungen in der <a href="/wiki/Algebraische_Zahlentheorie" title="Algebraische Zahlentheorie">algebraischen Zahlentheorie</a> (Harmonische Analysis auf <a href="/wiki/Adelring" class="mw-redirect" title="Adelring">Adelen</a>) und später das <a href="/wiki/Langlands-Programm" title="Langlands-Programm">Langlands-Programm</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=7" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=7" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Etingof, Golberg, Hensel, Liu, Schwendner, Vaintrob, Yudovina: <i>Introduction to Representation Theory</i>. AMS, 2011. <a href="/wiki/Spezial:ISBN-Suche/9780821853511" class="internal mw-magiclink-isbn">ISBN 978-0-8218-5351-1</a>.</li> <li>Roe Goodman, Nolan R. Wallach: <i>Symmetry, representations, and invariants.</i> (= <i>Graduate Texts in Mathematics.</i> 255). Springer, Dordrecht 2009, <a href="/wiki/Spezial:ISBN-Suche/9780387798516" class="internal mw-magiclink-isbn">ISBN 978-0-387-79851-6</a>.</li> <li>Brian C. Hall: <i>Lie groups, Lie algebras, and representations. An elementary introduction.</i> (= <i>Graduate Texts in Mathematics.</i> 222). Springer-Verlag, New York 2003, <a href="/wiki/Spezial:ISBN-Suche/0387401229" class="internal mw-magiclink-isbn">ISBN 0-387-40122-9</a>.</li> <li>Theodor Bröcker, Tammo tom Dieck: <i>Representations of compact Lie groups.</i> (= <i>Graduate Texts in Mathematics.</i> 98). Translated from the German manuscript. Corrected reprint of the 1985 translation. Springer-Verlag, New York 1995, <a href="/wiki/Spezial:ISBN-Suche/0387136789" class="internal mw-magiclink-isbn">ISBN 0-387-13678-9</a>.</li> <li>J. L. Alperin, Rowen B. Bell: <i>Groups and representations.</i> (= <i>Graduate Texts in Mathematics.</i> 162). Springer-Verlag, New York 1995, <a href="/wiki/Spezial:ISBN-Suche/0387945253" class="internal mw-magiclink-isbn">ISBN 0-387-94525-3</a>.</li> <li>William Fulton, Joe Harris: <i>Representation theory. A first course.</i> (= <i>Graduate Texts in Mathematics.</i> 129). Readings in Mathematics. Springer-Verlag, New York 1991, <a href="/wiki/Spezial:ISBN-Suche/0387975276" class="internal mw-magiclink-isbn">ISBN 0-387-97527-6</a>; 0-387-97495-4</li> <li>V. S. Varadarajan: <i>Lie groups, Lie algebras, and their representations.</i> (= <i>Graduate Texts in Mathematics.</i> 102). Reprint of the 1974 edition. Springer-Verlag, New York 1984, <a href="/wiki/Spezial:ISBN-Suche/0387909699" class="internal mw-magiclink-isbn">ISBN 0-387-90969-9</a>.</li> <li>James E. Humphreys: <i>Introduction to Lie algebras and representation theory.</i> (= <i>Graduate Texts in Mathematics.</i> 9). Second printing, revised. Springer-Verlag, New York / Berlin 1978, <a href="/wiki/Spezial:ISBN-Suche/0387900535" class="internal mw-magiclink-isbn">ISBN 0-387-90053-5</a>.</li> <li>Charles W. Curtis: <i>Pioneers of representation theory: Frobenius, Burnside, Schur, and Brauer.</i> (= <i>History of Mathematics.</i> 15). American Mathematical Society, Providence, RI/London Mathematical Society, London 1999, <a href="/wiki/Spezial:ISBN-Suche/0821890026" class="internal mw-magiclink-isbn">ISBN 0-8218-9002-6</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=8" title="Abschnitt bearbeiten: Weblinks" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=8" title="Quellcode des Abschnitts bearbeiten: Weblinks"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zur Geschichte der Darstellungstheorie: </p> <ul><li>Anthony W. Knapp: <i>Group representations and harmonic analysis from Euler to Langlands.</i> In: <i>Notices of the American Mathematical Society.</i> 43, 4, 1996, <a rel="nofollow" class="external text" href="https://www.math.stonybrook.edu/~aknapp/pdf-files/notices1.pdf">Teil 1</a>; 43, 5, 1996, <a rel="nofollow" class="external text" href="https://www.math.stonybrook.edu/~aknapp/pdf-files/notices2.pdf">Teil 2</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darstellungstheorie&veaction=edit&section=9" title="Abschnitt bearbeiten: Einzelnachweise" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Darstellungstheorie&action=edit&section=9" title="Quellcode des Abschnitts bearbeiten: Einzelnachweise"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Einleitung zu Knapp (<a href="/wiki/Op._cit." class="mw-redirect" title="Op. cit.">op. cit.</a>)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Teil 2 von Knapp (op. cit.)</span> </li> </ol> <div class="hintergrundfarbe1 rahmenfarbe1 navigation-not-searchable normdaten-typ-s" style="border-style: solid; border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;" id="normdaten"> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Normdaten (Sachbegriff): <a href="/wiki/Gemeinsame_Normdatei" title="Gemeinsame Normdatei">GND</a>: <span class="plainlinks-print"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4148816-7">4148816-7</a></span> <span class="noprint">(<a rel="nofollow" class="external text" href="https://lobid.org/gnd/4148816-7">lobid</a>, <a rel="nofollow" class="external text" href="https://swb.bsz-bw.de/DB=2.104/SET=1/TTL=1/CMD?retrace=0&trm_old=&ACT=SRCHA&IKT=2999&SRT=RLV&TRM=4148816-7">OGND</a><span class="metadata">, <a rel="nofollow" class="external text" href="https://prometheus.lmu.de/gnd/4148816-7">AKS</a></span>)</span> <span class="metadata"></span></div> </div></div></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Darstellungstheorie&oldid=249188353">https://de.wikipedia.org/w/index.php?title=Darstellungstheorie&oldid=249188353</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Darstellungstheorie" title="Kategorie:Darstellungstheorie">Darstellungstheorie</a></li><li><a href="/wiki/Kategorie:Teilgebiet_der_Mathematik" title="Kategorie:Teilgebiet der Mathematik">Teilgebiet der Mathematik</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Darstellungstheorie" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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