@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr: a skos:ConceptScheme .
psr:-KZLFZ6JQ-N
  skos:exactMatch <https://en.wikipedia.org/wiki/Poisson_manifold>, <https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Poisson> ;
  skos:prefLabel "variété de Poisson"@fr, "Poisson manifold"@en ;
  skos:inScheme psr: ;
  skos:narrower psr:-DTJ0P56T-7 ;
  skos:definition """En géométrie, une <b>structure de Poisson</b> sur une variété différentielle <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 M}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>M</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt=" M "></span> est un crochet de Lie <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 \\\\{\\\\cdot ,\\\\cdot \\\\}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo>,</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\{\\\\cdot ,\\\\cdot \\\\}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daac56121c7fdfb61a7c33d810be7487e0993460" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\\\\displaystyle \\\\{\\\\cdot ,\\\\cdot \\\\}}"></span> (appelé crochet de Poisson dans ce cas) sur l'algèbre <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 {C^{\\\\infty }}(M)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msup>
<br/>            <mi>C</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>            </mrow>
<br/>          </msup>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {C^{\\\\infty }}(M)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4f556427803c000d0ff61248491b81203842d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{\\\\displaystyle {C^{\\\\infty }}(M)}"></span> des fonctions lisses de <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 M}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>M</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt=" M "></span> à valeurs réelles, vérifiant formule de Leibniz
<br/>
<br/><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 \\\\{f,gh\\\\}=\\\\{f,g\\\\}h+g\\\\{f,h\\\\}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>f</mi>
<br/>        <mo>,</mo>
<br/>        <mi>g</mi>
<br/>        <mi>h</mi>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mo>=</mo>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>f</mi>
<br/>        <mo>,</mo>
<br/>        <mi>g</mi>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mi>h</mi>
<br/>        <mo>+</mo>
<br/>        <mi>g</mi>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>f</mi>
<br/>        <mo>,</mo>
<br/>        <mi>h</mi>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\{f,gh\\\\}=\\\\{f,g\\\\}h+g\\\\{f,h\\\\}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c34874254fd5d534cd4f2b6e0ea2b4e284ada785" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:27.216ex; height:2.843ex;" alt="{\\\\displaystyle \\\\{f,gh\\\\}=\\\\{f,g\\\\}h+g\\\\{f,h\\\\}}"></span>.</dd></dl>
<br/>En d'autres termes, une structure de Poisson  est structure d'algèbre de Lie sur l'espace vectoriel des fonctions lisses sur <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 M}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>M</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt=" M "></span> de sorte que <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 X_{f}{\\\\stackrel {\\	ext{df}}{=}}\\\\{f,\\\\cdot \\\\}:{C^{\\\\infty }}(M)\\	o {C^{\\\\infty }}(M)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>X</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>f</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow class="MJX-TeXAtom-REL">
<br/>            <mover>
<br/>              <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<br/>                <mo>=</mo>
<br/>              </mrow>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mtext>df</mtext>
<br/>              </mrow>
<br/>            </mover>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mi>f</mi>
<br/>        <mo>,</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mo>:</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msup>
<br/>            <mi>C</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>            </mrow>
<br/>          </msup>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msup>
<br/>            <mi>C</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>            </mrow>
<br/>          </msup>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle X_{f}{\\\\stackrel {\\	ext{df}}{=}}\\\\{f,\\\\cdot \\\\}:{C^{\\\\infty }}(M)\\	o {C^{\\\\infty }}(M)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d09930fa191ede90647f3d88fd8ce1e416b08a4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:31.554ex; height:4.009ex;" alt="{\\\\displaystyle X_{f}{\\\\stackrel {\\	ext{df}}{=}}\\\\{f,\\\\cdot \\\\}:{C^{\\\\infty }}(M)\\	o {C^{\\\\infty }}(M)}"></span> est un champ de vecteurs pour toute fonction lisse  <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 f}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt=" f "></span>, appelé champ de vecteurs hamiltonien associé à <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 f}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>f</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle f}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt=" f "></span>. 
<br/> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Poisson">https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_de_Poisson</a>)"""@fr, """In differential geometry, a field in mathematics, a <b>Poisson manifold</b> is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in turn generalises the phase space from Hamiltonian mechanics.
<br/>A <b>Poisson structure</b> (or Poisson bracket) on a smooth manifold <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 M}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>M</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="M"></span> is a function<div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\{\\\\cdot ,\\\\cdot \\\\}:{\\\\mathcal {C}}^{\\\\infty }(M)\\	imes {\\\\mathcal {C}}^{\\\\infty }(M)\\	o {\\\\mathcal {C}}^{\\\\infty }(M)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo fence="false" stretchy="false">{</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo>,</mo>
<br/>        <mo>⋅<!-- ⋅ --></mo>
<br/>        <mo fence="false" stretchy="false">}</mo>
<br/>        <mo>:</mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>×<!-- × --></mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <msup>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>          </mrow>
<br/>        </msup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\{\\\\cdot ,\\\\cdot \\\\}:{\\\\mathcal {C}}^{\\\\infty }(M)\\	imes {\\\\mathcal {C}}^{\\\\infty }(M)\\	o {\\\\mathcal {C}}^{\\\\infty }(M)}</annotation>
<br/>  </semantics>
<br/></math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7cb6865f582653114a181ed49b545dd9805b79b" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:35.155ex; height:2.843ex;" alt="{\\\\displaystyle \\\\{\\\\cdot ,\\\\cdot \\\\}:{\\\\mathcal {C}}^{\\\\infty }(M)\\	imes {\\\\mathcal {C}}^{\\\\infty }(M)\\	o {\\\\mathcal {C}}^{\\\\infty }(M)}"></div>on the vector space <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 {C^{\\\\infty }}(M)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <msup>
<br/>            <mi>C</mi>
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">∞<!-- ∞ --></mi>
<br/>            </mrow>
<br/>          </msup>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>M</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {C^{\\\\infty }}(M)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a4f556427803c000d0ff61248491b81203842d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:7.925ex; height:2.843ex;" alt="{C^{{\\\\infty }}}(M)"></span> of smooth functions on <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 M}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>M</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle M}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="M"></span>, making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra).
<br/>Poisson structures on manifolds were introduced by André Lichnerowicz in 1977 and are named after the French mathematician Siméon Denis Poisson, due to their early appearance in his works on analytical mechanics.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Poisson_manifold">https://en.wikipedia.org/wiki/Poisson_manifold</a>)"""@en ;
  a skos:Concept ;
  skos:broader psr:-RZMJ5VH2-S .

psr:-DTJ0P56T-7
  skos:prefLabel "Poisson-Lie group"@en, "groupe de Lie-Poisson"@fr ;
  a skos:Concept ;
  skos:broader psr:-KZLFZ6JQ-N .

psr:-RZMJ5VH2-S
  skos:prefLabel "differentiable manifold"@en, "variété différentielle"@fr ;
  a skos:Concept ;
  skos:narrower psr:-KZLFZ6JQ-N .