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<title>Poisson ring</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Poisson ring</span></span>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>Poisson ring</b> is a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> on which an <a href="Anticommutativity" class="mw-redirect" title="Anticommutativity">anticommutative</a> and <a href="Distributivity" class="mw-redirect" title="Distributivity">distributive</a> <a href="Binary_operation" title="Binary operation">binary operation</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 [\cdot ,\cdot ]}">

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

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</math></span><img src="./28dd4c22d60192519c1c12cf645b040f368db9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [\cdot ,\cdot ]}" loading="lazy"></span> satisfying the <a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a> and the <a href="Product_rule" title="Product rule">product rule</a> is defined. Such an operation is then known as the <a href="Poisson_bracket" title="Poisson bracket">Poisson bracket</a> of the Poisson ring.

</p><p>Many important operations and results of <a href="Symplectic_geometry" title="Symplectic geometry">symplectic geometry</a> and <a href="Hamiltonian_mechanics" title="Hamiltonian mechanics">Hamiltonian mechanics</a> may be formulated in terms of the Poisson bracket and, hence, apply to <a href="Poisson_algebra" title="Poisson algebra">Poisson algebras</a> as well. This observation is important in studying the <a href="Classical_limit" title="Classical limit">classical limit</a> of <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>—the <a href="Non-commutative_algebra" class="mw-redirect" title="Non-commutative algebra">non-commutative algebra</a> of <a href="Operator_(mathematics)" title="Operator (mathematics)">operators</a> on a <a href="Hilbert_space" title="Hilbert space">Hilbert space</a> has the Poisson algebra of functions on a <a href="Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.

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

<p>The Poisson bracket must satisfy the identities

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

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</math></span><img src="./0fca41d790d01ae352035e6a15680c369ffe695a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.351ex; height:2.843ex;" alt="{\displaystyle [f,g]=-[g,f]}" loading="lazy"></span> (skew symmetry)</li>

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

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</math></span><img src="./d14d383916d3fa58ff33908769b1a7e19faa9cc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.568ex; height:2.843ex;" alt="{\displaystyle [f+g,h]=[f,h]+[g,h]}" loading="lazy"></span> (distributivity)</li>

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

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</math></span><img src="./1b5d39004590742d3349958710d6329226221541.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.122ex; height:2.843ex;" alt="{\displaystyle [fg,h]=f[g,h]+[f,h]g}" loading="lazy"></span> (<a href="Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a>)</li>

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<annotation encoding="application/x-tex">{\displaystyle [f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0}</annotation>

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</math></span><img src="./f16d4100f7df5af9370ab0ef4c0301145b2d9a60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.108ex; height:2.843ex;" alt="{\displaystyle [f,[g,h]]+[g,[h,f]]+[h,[f,g]]=0}" loading="lazy"></span> (<a href="Jacobi_identity" title="Jacobi identity">Jacobi identity</a>)</li></ul>

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

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</math></span><img src="./40bac5cdee81bb0c899b395c37f6ab3e3a8a0193.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.801ex; height:2.509ex;" alt="{\displaystyle f,g,h}" loading="lazy"></span> in the ring.

</p><p>A <a href="Poisson_algebra" title="Poisson algebra">Poisson algebra</a> is a Poisson ring that is also an <a href="Algebra_over_a_field" title="Algebra over a field">algebra over a field</a>. In this case, add the extra requirement

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

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</math></span><img src="./8bd1e119ea92e2d680c793fc60ea636c6d4a2daf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.724ex; height:2.843ex;" alt="{\displaystyle [sf,g]=s[f,g]}" loading="lazy"></span></dd></dl>

<p>for all scalars <i>s</i>.

</p><p>For each <i>g</i> in a Poisson ring <i>A</i>, the operation <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 ad_{g}}">

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

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</math></span><img src="./90d19aaf12686c9b192da236c57a41b9d8ae0ebc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.46ex; height:2.843ex;" alt="{\displaystyle ad_{g}}" loading="lazy"></span> defined as <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 ad_{g}(f)=[f,g]}">

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<annotation encoding="application/x-tex">{\displaystyle ad_{g}(f)=[f,g]}</annotation>

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</math></span><img src="./c0742abfb792d57431a194a671b1861e7843dc45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.369ex; height:3.009ex;" alt="{\displaystyle ad_{g}(f)=[f,g]}" loading="lazy"></span> is a <a href="Derivation_(abstract_algebra)" class="mw-redirect" title="Derivation (abstract algebra)">derivation</a>. If the set <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 \{ad_{g}|g\in A\}}">

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

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</math></span><img src="./0a488b46db90fc7426d9c97214abd3ec3a2398ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.132ex; height:3.009ex;" alt="{\displaystyle \{ad_{g}|g\in A\}}" loading="lazy"></span> generates the set of derivations of <i>A</i>, then <i>A</i> is said to be <b>non-degenerate</b>.

</p><p>If a non-degenerate Poisson ring is <a href="Ring_isomorphism" class="mw-redirect" title="Ring isomorphism">isomorphic as a commutative ring</a> to the algebra of smooth functions on a manifold <i>M</i>, then <i>M</i> must be a <a href="Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> and <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 ,\cdot ]}">

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

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</math></span><img src="./28dd4c22d60192519c1c12cf645b040f368db9e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.621ex; height:2.843ex;" alt="{\displaystyle [\cdot ,\cdot ]}" loading="lazy"></span> is the Poisson bracket defined by the <a href="Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a>.

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://planetmath.org/IfTheAlgebraOfFunctionsOnAManifoldIsAPoissonRingThenTheManifoldIsSymplectic">"If the algebra of functions on a manifold is a Poisson ring then the manifold is symplectic"</a>. <i><a href="PlanetMath" title="PlanetMath">PlanetMath</a></i>.</cite></li></ul>

<p><i>This article incorporates material from Poisson Ring on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>

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This article is issued from <a class="external text" title="Last edited on 2022-11-27" href="https://en.wikipedia.org/wiki/?title=Poisson_ring&oldid=1124164093">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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