GermanWikipedia - /page/entry.mu

<!DOCTYPE html>

<title>Poisson-Algebra</title>

</head>

<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Poisson-Algebra</span></h1>

</header>

</div>

<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>Eine <b>Poisson-Algebra</b> ist in der <a href="Mathematik" title="Mathematik">Mathematik</a> eine <a href="Kommutativgesetz" title="Kommutativgesetz">kommutative</a>, <a href="Assoziative_Algebra" title="Assoziative Algebra">assoziative Algebra</a>, welche mit einer <a href="Poisson-Klammer" title="Poisson-Klammer">Poisson-Klammer</a> ausgestattet ist. Die Klammer ist eine <a href="Lie-Klammer" title="Lie-Klammer">Lie-Klammer</a>, welche zusätzlich die <a href="Produktregel" title="Produktregel">Leibnizregel</a> erfüllt, das heißt sie ist eine <a href="Derivation_(Mathematik)" title="Derivation (Mathematik)">Derivation</a> der assoziativen Multiplikation.

</p>

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

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"></span> ein <a href="Kommutativer_Ring" class="mw-redirect" title="Kommutativer Ring">kommutativer Ring</a>. Eine <b>Poisson-Algebra</b> <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}">

</mstyle>

</mrow>

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

</semantics>

</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> ist eine kommutative, assoziative <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 R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"></span>-Algebra <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,\cdot )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A,\cdot )}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ac43a0ba7d67fce39388e04b490963050f097d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.233ex; height:2.843ex;" alt="{\displaystyle (A,\cdot )}" loading="lazy"></span> mit einer <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 R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"></span>-<a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilinearen</a> und <a href="Bilinearform#Symmetrieeigenschaften_im_Fall_V_=_W" title="Bilinearform">antisymmetrischen</a> Abbildung

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{-,-\}:A\times A\to A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9f140164b29983485e10cf123fae526d684bfc5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.596ex; height:2.843ex;" alt="{\displaystyle \{-,-\}:A\times A\to A}" loading="lazy"></span>,</dd></dl>

<p>genannt <i>Poisson-Klammer</i>, so dass

</p>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/58bc0083cf85bf5db7b7d4315181e8afefd1c8b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.561ex; height:2.843ex;" alt="{\displaystyle (A,\{-,-\})}" loading="lazy"></span> eine <a href="Lie-Algebra" title="Lie-Algebra">Lie-Algebra</a> über <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 R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"></span> ist,</li>

<li>die Poisson-Klammer die Leibnizregel erfüllt</li></ul>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a1d8dd1efb99f81749748b4e04c7d49f87aff51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.253ex; height:2.843ex;" alt="{\displaystyle \{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}}" loading="lazy"></span>.<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></dd></dl></dd></dl>

<p>Die Striche in der leeren Poisson-Klammer <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 \{-,-\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cda2dc5437fe9b2c14f96108071c0e3ba178f32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle \{-,-\}}" loading="lazy"></span> stehen dabei für einen Platzhalter.

</p>

<div class="mw-heading mw-heading3"><h3 id="Erläuterungen"><span id="Erl.C3.A4uterungen"></span>Erläuterungen</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"></span>-<a href="Modul_(Mathematik)" title="Modul (Mathematik)">Modul</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}">

</mstyle>

</mrow>

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

</semantics>

</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> ist mit zwei <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 R}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.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 R}" loading="lazy"></span>-bilinearen Abbildungen ausgestattet, der Multiplikation <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 \cdot \colon A\times A\to A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot \colon A\times A\to A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/80ecc37a4aae32f23d5d8f3de3bb8ed77133531f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.365ex; height:2.176ex;" alt="{\displaystyle \cdot \colon A\times A\to A}" loading="lazy"></span> und der Poisson-Klammer <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 \{-,-\}\colon A\times A\to A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{-,-\}\colon A\times A\to A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f68de49cda450e372a694df7da743f69ba679d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.693ex; height:2.843ex;" alt="{\displaystyle \{-,-\}\colon A\times A\to A}" loading="lazy"></span>.

</p><p>Für die Multiplikation <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 \cdot }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></span> und <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 f,g,h\in A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g,h\in A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c36915a10f5fed295739589e0154c7ab35f7bce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.385ex; height:2.509ex;" alt="{\displaystyle f,g,h\in A}" loading="lazy"></span> gilt

</p>

<dl><dd>Kommutativität: <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 f\cdot g=g\cdot f}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/427692478806c24ed2f61ed7a19e128f9056d5de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.246ex; height:2.509ex;" alt="{\displaystyle f\cdot g=g\cdot f}" loading="lazy"></span></dd>

<dd>Assoziativität: <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 f\cdot (g\cdot h)=(f\cdot g)\cdot h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\cdot (g\cdot h)=(f\cdot g)\cdot h}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0074ad95649d6dc755fd074016eb671ded30c3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.901ex; height:2.843ex;" alt="{\displaystyle f\cdot (g\cdot h)=(f\cdot g)\cdot h}" loading="lazy"></span></dd></dl>

<p>Für die Poisson-Klammer <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 \{-,-\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cda2dc5437fe9b2c14f96108071c0e3ba178f32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle \{-,-\}}" loading="lazy"></span> und <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 f,g,h\in A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g,h\in A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c36915a10f5fed295739589e0154c7ab35f7bce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.385ex; height:2.509ex;" alt="{\displaystyle f,g,h\in A}" loading="lazy"></span> gilt

</p>

<dl><dd>Antisymmetrie: <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 \{f,g\}=-\{g,f\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eff1f1f4a7eef368856844d55a700e559463abe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.413ex; height:2.843ex;" alt="{\displaystyle \{f,g\}=-\{g,f\}}" loading="lazy"></span> und <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 \{f,f\}=0}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9091706a9a5bfcae472e4565cb18c7f272f6982d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.177ex; height:2.843ex;" alt="{\displaystyle \{f,f\}=0}" loading="lazy"></span></dd>

<dd>Leibnizregel: <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 \{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a1d8dd1efb99f81749748b4e04c7d49f87aff51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.253ex; height:2.843ex;" alt="{\displaystyle \{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}}" loading="lazy"></span></dd>

<dd><a href="Jacobi-Identit%C3%A4t" title="Jacobi-Identität">Jacobi-Identität</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 \{f,\{g,h\}\}=\{\{f,g\},h\}+\{g,\{f,h\}\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{f,\{g,h\}\}=\{\{f,g\},h\}+\{g,\{f,h\}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60edc5b01c328b2c76549befae11aaa773355e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.293ex; height:2.843ex;" alt="{\displaystyle \{f,\{g,h\}\}=\{\{f,g\},h\}+\{g,\{f,h\}\}}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65aa8d2f370727a799a1c413e553afad12518189.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.862ex; height:2.509ex;" alt="{\displaystyle f\in A}" loading="lazy"></span> ist die Poisson-Klammer <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 D_{f}(-):=\{f,-\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}(-):=\{f,-\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc38cbbebdbf2107c49f006175bed8278019e516.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.869ex; height:3.009ex;" alt="{\displaystyle D_{f}(-):=\{f,-\}}" loading="lazy"></span> eine Derivation der Multiplikation, denn es gilt nach den Regeln

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{f}(g\cdot h)=\{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}=D_{f}(g)\cdot h+g\cdot D_{f}(h).}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98775d6a756d290e86099329e7946e86010c0006.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:68.95ex; height:3.009ex;" alt="{\displaystyle D_{f}(g\cdot h)=\{f,g\cdot h\}=\{f,g\}\cdot h+g\cdot \{f,h\}=D_{f}(g)\cdot h+g\cdot D_{f}(h).}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Poisson-*-Algebra"><span id="Poisson-.2A-Algebra"></span>Poisson-*-Algebra</h3></div>

<p>Falls <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}">

</mstyle>

</mrow>

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

</semantics>

</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> eine Poisson-Algebra über <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 \mathbb {C} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9add4085095b9b6d28d045fd9c92c2c09f549a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }" loading="lazy"></span> ist, die zusätzlich eine <a href="*-Algebra" title="*-Algebra">*-Algebra</a> ist und für <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 \{-,-\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cda2dc5437fe9b2c14f96108071c0e3ba178f32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle \{-,-\}}" loading="lazy"></span> folgendes

</p>

<mover>

<mrow>

</mrow>

</mover>

</mrow>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64441f797197052a1f3afd1065b429a90d1c90df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.045ex; height:3.676ex;" alt="{\displaystyle {\overline {\{f,g\}}}=\{{\overline {f}},{\overline {g}}\}}" loading="lazy"></span></dd></dl>

<p>erfüllt, so nennt man <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}">

</mstyle>

</mrow>

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

</semantics>

</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> eine <b>Poisson-*-Algebra</b>.<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="Beispiele">Beispiele</h2></div>

</mstyle>

</mrow>

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

</semantics>

</math></span><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"></span> eine <a href="Poisson-Mannigfaltigkeit" title="Poisson-Mannigfaltigkeit">Poisson-Mannigfaltigkeit</a> mit der Poisson-Klammer <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 \{-,-\}}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cda2dc5437fe9b2c14f96108071c0e3ba178f32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.975ex; height:2.843ex;" alt="{\displaystyle \{-,-\}}" loading="lazy"></span> auf dem Raum der <a href="Glatte_Funktion" title="Glatte Funktion">glatten Funktionen</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 C^{\infty }(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C^{\infty }(M)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48b543485314a4426ee2ff956f206cfd802d68d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\displaystyle C^{\infty }(M)}" loading="lazy"></span>, dann ist das Paar <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 (C^{\infty }(M),\{-,-\})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (C^{\infty }(M),\{-,-\})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6effbefda7cfdf4d686b0b1f6769691c0a319c01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.743ex; height:2.843ex;" alt="{\displaystyle (C^{\infty }(M),\{-,-\})}" loading="lazy"></span> eine Poisson-Algebra.</li></ul>

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

<ul><li>Stefan Waldmann: <cite style="font-style:italic">Poisson-Geometrie und Deformationsquantisierung</cite>. Springer Verlag, 2001, ISBN 978-3-540-72517-6.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Poisson-Algebra&rft.au=Stefan+Waldmann&rft.btitle=Poisson-Geometrie+und+Deformationsquantisierung&rft.date=2001&rft.genre=book&rft.isbn=9783540725176&rft.pub=Springer+Verlag" style="display:none"> </span></li>

<li>Chiara Esposito: <cite style="font-style:italic">Formality Theory</cite>. Springer Verlag, 2015, ISBN 978-3-319-09289-8.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Poisson-Algebra&rft.au=Chiara+Esposito&rft.btitle=Formality+Theory&rft.date=2015&rft.genre=book&rft.isbn=9783319092898&rft.pub=Springer+Verlag" style="display:none"> </span></li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/index.php?title=Poisson_algebra">Poisson-Algebra in der Encyclopedia of Mathematics</a></li></ul>

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

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Chiara Esposito: <cite style="font-style:italic">Formality Theory</cite>. Springer Verlag, 2015, ISBN 978-3-319-09289-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>10–11</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Poisson-Algebra&rft.au=Chiara+Esposito&rft.btitle=Formality+Theory&rft.date=2015&rft.genre=book&rft.isbn=9783319092898&rft.pages=10-11&rft.pub=Springer+Verlag" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Stefan Waldmann: <cite style="font-style:italic">Poisson-Geometrie und Deformationsquantisierung</cite>. Springer Verlag, 2001, ISBN 978-3-540-72517-6, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>20</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Poisson-Algebra&rft.au=Stefan+Waldmann&rft.btitle=Poisson-Geometrie+und+Deformationsquantisierung&rft.date=2001&rft.genre=book&rft.isbn=9783540725176&rft.pages=20&rft.pub=Springer+Verlag" style="display:none"> </span></span>

</li>

</ol></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-01-08" href="https://de.wikipedia.org/wiki/?title=Poisson-Algebra&oldid=240984010">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.

</div>

</div></div>

</div>

</main>

</div>

</body></html>