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<title>Killing-Form</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Killing-Form</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">Die Killing-Form (auch Cartan-Killing-Form) spielt eine wichtige Rolle in der <a href="Differentialgeometrie" title="Differentialgeometrie">Differentialgeometrie</a> und in der Klassifikation der <a href="Halbeinfache_Lie-Algebra" title="Halbeinfache Lie-Algebra">halbeinfachen Lie-Algebren</a>.

Sie ist nach <a href="Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a> benannt.

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> eine <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a> über dem Körper <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} :{\mathfrak {g}}\rightarrow {\mathfrak {gl}}({\mathfrak {g}})}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {ad} :{\mathfrak {g}}\rightarrow {\mathfrak {gl}}({\mathfrak {g}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8b8a4c9a117e3d43624be851de705dee9ce7f1f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.982ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} :{\mathfrak {g}}\rightarrow {\mathfrak {gl}}({\mathfrak {g}})}" loading="lazy"> ihre <a href="Adjungierte_Darstellung" title="Adjungierte Darstellung">adjungierte Darstellung</a>.

Die Killing-Form ist die durch

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<annotation encoding="application/x-tex">{\displaystyle B(X,Y):=\operatorname {Tr} (\operatorname {ad} (X)\circ \operatorname {ad} (Y))}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4a04678ae268adb9084474db73c69f2a37635507.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.981ex; height:2.843ex;" alt="{\displaystyle B(X,Y):=\operatorname {Tr} (\operatorname {ad} (X)\circ \operatorname {ad} (Y))}" loading="lazy"></dd></dl>

für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\in {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle X,Y\in {\mathfrak {g}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10acea4f450fba3f400cd9e0a6c1ef76a0359a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.8ex; height:2.509ex;" alt="{\displaystyle X,Y\in {\mathfrak {g}}}" loading="lazy"> definierte <a href="Symmetrische_Bilinearform" class="mw-redirect" title="Symmetrische Bilinearform">symmetrische Bilinearform</a>

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<annotation encoding="application/x-tex">{\displaystyle B:{\mathfrak {g}}\times {\mathfrak {g}}\rightarrow k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5115547ac9969f64b39ad2b0a5ec8e6935180d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.71ex; height:2.509ex;" alt="{\displaystyle B:{\mathfrak {g}}\times {\mathfrak {g}}\rightarrow k}" loading="lazy">,</dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37afaa40330d0e6a3eff1767cf6bad007f56412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.59ex; height:2.176ex;" alt="{\displaystyle \operatorname {Tr} }" loading="lazy"> die <a href="Spur_(Mathematik)" title="Spur (Mathematik)">Spur</a> bezeichnet.

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ist eine symmetrische Bilinearform.</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> ist assoziativ, das heißt, es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B([X,Y],Z)=B(X,[Y,Z])}">

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<annotation encoding="application/x-tex">{\displaystyle B([X,Y],Z)=B(X,[Y,Z])}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0aed22314d31dbeac4c8ba761a71873d77b67ebb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.836ex; height:2.843ex;" alt="{\displaystyle B([X,Y],Z)=B(X,[Y,Z])}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y,Z\in {\mathfrak {g}}}">

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<annotation encoding="application/x-tex">{\displaystyle X,Y,Z\in {\mathfrak {g}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61c961731da93c32f687676e38bfda6a02ad6f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.514ex; height:2.509ex;" alt="{\displaystyle X,Y,Z\in {\mathfrak {g}}}" loading="lazy">.</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18d5864036c9eb31a1f9334b3185bdd1f9cc553d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.693ex; height:2.509ex;" alt="{\displaystyle Z\in {\mathfrak {g}}}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {ad} (Z)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/89d679b546d1444bd8be98907a091add0a38928a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.945ex; height:2.843ex;" alt="{\displaystyle \operatorname {ad} (Z)}" loading="lazy"> schiefsymmetrisch bzgl. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy">, das heißt für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,Y\in {\mathfrak {g}}}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,Y\in {\mathfrak {g}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10acea4f450fba3f400cd9e0a6c1ef76a0359a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.8ex; height:2.509ex;" alt="{\displaystyle X,Y\in {\mathfrak {g}}}" loading="lazy"> gilt</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle B(\operatorname {ad} (Z)X,Y)=-B(X,\operatorname {ad} (Z)Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe03b71ceb146f9616f2f5ce68c62c7b0fca920f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.517ex; height:2.843ex;" alt="{\displaystyle B(\operatorname {ad} (Z)X,Y)=-B(X,\operatorname {ad} (Z)Y)}" loading="lazy">.</dd></dl>

<ul><li>Die Killing-Form ist <a href="Bilinearform#Nicht_ausgeartete_Bilinearform" title="Bilinearform">nicht-ausgeartet</a> genau dann, wenn die Lie-Algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> halb-einfach ist.</li>

<li>Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> die Lie-Algebra einer <a href="Lie-Gruppe" title="Lie-Gruppe">Lie-Gruppe</a> <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="./_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"> ist, dann ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ad} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32802447373caef9cd6d7ce1acfe0c1552d4bf98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.036ex; height:2.176ex;" alt="{\displaystyle \operatorname {Ad} }" loading="lazy">-invariant, d. h. für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G,X,Y\in {\mathfrak {g}}}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G,X,Y\in {\mathfrak {g}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/852b1dc49c4de5b118899cdd13c537cf8736090b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.617ex; height:2.509ex;" alt="{\displaystyle g\in G,X,Y\in {\mathfrak {g}}}" loading="lazy"> gilt</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle B(\operatorname {Ad} (g)X,\operatorname {Ad} (g)Y)=B(X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b3cbabc3fde434a90bb9d7a2c26bcd43ebe3b8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.741ex; height:2.843ex;" alt="{\displaystyle B(\operatorname {Ad} (g)X,\operatorname {Ad} (g)Y)=B(X,Y)}" loading="lazy">.</dd></dl>

<ul><li>Falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40a913b1503ed9ec94361b99f7fd59ef60705c28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}" loading="lazy"> die Lie-Algebra einer <a href="Halbeinfache_Lie-Gruppe" title="Halbeinfache Lie-Gruppe">halbeinfachen Lie-Gruppe</a> ist, dann ist die Killing-Form <a href="Negativ_definit" class="mw-redirect" title="Negativ definit">negativ definit</a> genau dann, wenn <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="./_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"> kompakt ist. Insbesondere definiert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ac21f1abfa9648a95f4b9f18063d78811b6dd96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.572ex; height:2.343ex;" alt="{\displaystyle -B}" loading="lazy"> eine bi-invariante <a href="Riemannsche_Metrik" class="mw-redirect" title="Riemannsche Metrik">Riemannsche Metrik</a> auf einer kompakten, halbeinfachen Lie-Gruppe <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="./_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">. Allgemeiner ist auf der Lie-Algebra einer kompakten (nicht notwendig halbeinfachen) Lie-Gruppe die Killingform stets <a href="Negativ_semidefinit" class="mw-redirect" title="Negativ semidefinit">negativ semidefinit</a>.</li></ul>

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

Die Killing-Form <a href="Nilpotente_Lie-Algebra" class="mw-redirect" title="Nilpotente Lie-Algebra">nilpotenter Lie-Algebren</a> ist identisch Null.

Für viele klassische Lie-Algebren lässt sich die Killing-Form explizit angeben:

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<annotation encoding="application/x-tex">{\displaystyle B(X,Y)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/161c8cf2a741eaf454166d4c6bf263ff1420d457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.361ex; height:2.843ex;" alt="{\displaystyle B(X,Y)}" loading="lazy">

</th></tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4d2e1b837ec31c9430c86cc088ac1c22ac8f50be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.425ex; height:2.843ex;" alt="{\displaystyle 2n\operatorname {Tr} (XY)-2\operatorname {Tr} (X)\operatorname {Tr} (Y)}" loading="lazy">

</td></tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb859fdaf2d57a3e0fceafa865532c2c48f59e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.097ex; height:2.843ex;" alt="{\displaystyle 2n\operatorname {Tr} (XY)}" loading="lazy">

</td></tr>

<tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb859fdaf2d57a3e0fceafa865532c2c48f59e02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.097ex; height:2.843ex;" alt="{\displaystyle 2n\operatorname {Tr} (XY)}" loading="lazy">

</td></tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8e0621795f6c3eff7e4f4dc293a0cb303b226dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.746ex; height:2.843ex;" alt="{\displaystyle (n-2)\operatorname {Tr} (XY)}" loading="lazy">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8e0621795f6c3eff7e4f4dc293a0cb303b226dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.746ex; height:2.843ex;" alt="{\displaystyle (n-2)\operatorname {Tr} (XY)}" loading="lazy">

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<tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e887299ae5ed480c74e4a6993a2edf567072b9ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.909ex; height:2.843ex;" alt="{\displaystyle (2n+2)\operatorname {Tr} (XY)}" loading="lazy">

</td></tr>

<tr>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e887299ae5ed480c74e4a6993a2edf567072b9ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.909ex; height:2.843ex;" alt="{\displaystyle (2n+2)\operatorname {Tr} (XY)}" loading="lazy">

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<div class="mw-heading mw-heading2"><h2 id="Riemannsche_Metrik_auf_symmetrischen_Räumen_von_nichtkompaktem_Typ">Riemannsche Metrik auf symmetrischen Räumen von nichtkompaktem Typ</h2></div>

Ein <a href="Symmetrischer_Raum" title="Symmetrischer Raum">symmetrischer Raum</a> von nichtkompaktem Typ ist eine Mannigfaltigkeit der Form

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd7c72070143c9b1c126f0bec454fc8df49cf0be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.596ex; height:2.843ex;" alt="{\displaystyle M=G/K}" loading="lazy"></dd></dl>

mit einer <a href="Halbeinfache_Lie-Gruppe" title="Halbeinfache Lie-Gruppe">halbeinfachen Lie-Gruppe</a> <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="./_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"> und einer maximal kompakten Untergruppe <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.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">.

Zu einem symmetrischen Raum hat man eine <a href="Cartan-Zerlegung" class="mw-redirect" title="Cartan-Zerlegung">Cartan-Zerlegung</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/578c1225c7847477ccbafa7548bb88f3b3778fc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.178ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}}}" loading="lazy"></dd></dl>

und man kann den Tangentialraum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\left[e\right]}G/K}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{\left[e\right]}G/K}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2bfcb6c1c1f9259b96dba23cc3c27aa198acb21e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.326ex; height:3.176ex;" alt="{\displaystyle T_{\left[e\right]}G/K}" loading="lazy"> im neutralen Element mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> identifizieren.

Die Killing-Form ist negativ definit auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {k}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/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"> und positiv definit auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">. Insbesondere definiert sie ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ad} (G)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/baff96305e5aff1261693de479e6fa52cfe7d0f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.672ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ad} (G)}" loading="lazy">-invariantes Skalarprodukt auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> und damit eine links-invariante Riemannsche Metrik auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=G/K}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd7c72070143c9b1c126f0bec454fc8df49cf0be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.596ex; height:2.843ex;" alt="{\displaystyle M=G/K}" loading="lazy">. Bis auf Multiplikation mit Skalaren ist dies die einzige <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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

</semantics>

</math><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">-invariante Metrik auf <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="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">.

Die Differentialgeometrie symmetrischer Räume beschäftigt sich mit den Eigenschaften dieser Riemannschen Mannigfaltigkeiten.

<div class="mw-heading mw-heading2"><h2 id="Klassifikation_halbeinfacher_Lie-Algebren">Klassifikation halbeinfacher Lie-Algebren</h2></div>

Die Killing-Form spielt eine Schlüsselrolle in der <a href="Halbeinfache_Lie-Algebra" title="Halbeinfache Lie-Algebra">Klassifikation der halbeinfachen Lie-Algebren</a> über <a href="Algebraisch_abgeschlossen" class="mw-redirect" title="Algebraisch abgeschlossen">algebraisch abgeschlossenen</a> Körpern der Charakteristik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">.

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

<ul><li>Humphreys, James E.: Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, Vol. 9. Springer-Verlag, New York-Berlin, 1972.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2020-01-17" href="https://de.wikipedia.org/wiki/?title=Killing-Form&oldid=195906990">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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