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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Abelsche Lie-Algebra</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Abelsche Lie-Algebren</b> sind ein Begriff aus der <a href="Mathematik" title="Mathematik">mathematischen</a> Theorie der <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppen</a> und <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebren</a>.
</p><p>Eine Lie-Algebra ist <i>abelsch</i>, wenn die Lie-Klammer identisch null ist.
</p><p>Jeder Vektorraum bildet eine abelsche Lie-Algebra, wenn man jede Lie-Klammer als Null definiert.
</p><p>Wenn die <a href="Lie-Gruppe#Lie-Algebra_der_Lie-Gruppe" title="Lie-Gruppe">Lie-Algebra der Lie-Gruppe</a> <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>G</mi>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"></span> eine abelsche Lie-Algebra ist, dann lässt sich <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>G</mi>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"></span> als <a href="Semidirektes_Produkt" title="Semidirektes Produkt">semidirektes Produkt</a>
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<dl><dd><span class="mwe-math-element mwe-math-element-inline"><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=A\rtimes \Gamma }">
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<annotation encoding="application/x-tex">{\displaystyle G=A\rtimes \Gamma }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f1652b2f435cc4461e85fefc3b182914d04dc43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.961ex; height:2.176ex;" alt="{\displaystyle G=A\rtimes \Gamma }" loading="lazy"></span></dd></dl>
<p>aus einer <a href="Abelsche_Lie-Gruppe" title="Abelsche Lie-Gruppe">abelschen Lie-Gruppe</a> <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>A</mi>
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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.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 A}" loading="lazy"></span> und einer <a href="Diskrete_Topologie" title="Diskrete Topologie">diskreten</a> Gruppe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }">
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<mi mathvariant="normal">Γ<!-- Γ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.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"></span> zerlegen.
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