@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-RMQ1RP9W-P
  skos:prefLabel "groupe de Lie"@fr, "Lie group"@en ;
  a skos:Concept ;
  skos:narrower psr:-K130R6C6-4 .

psr:-ZB0DDWPX-0
  skos:prefLabel "représentation thêta"@fr, "theta representation"@en ;
  a skos:Concept ;
  skos:broader psr:-K130R6C6-4 .

psr: a skos:ConceptScheme .
psr:-QDSZ76FR-B
  skos:prefLabel "quantum field theory"@en, "théorie quantique des champs"@fr ;
  a skos:Concept ;
  skos:narrower psr:-K130R6C6-4 .

psr:-P43HJWNV-X
  skos:prefLabel "symplectic geometry"@en, "géométrie symplectique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-K130R6C6-4 .

psr:-K130R6C6-4
  skos:exactMatch <https://en.wikipedia.org/wiki/Heisenberg_group>, <https://fr.wikipedia.org/wiki/Groupe_de_Heisenberg> ;
  skos:prefLabel "Heisenberg group"@en, "groupe de Heisenberg"@fr ;
  skos:narrower psr:-ZB0DDWPX-0 ;
  skos:definition """In mathematics, the <b>Heisenberg group</b> <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 H}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>H</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle H}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="H"></span>, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
<br/>
<br/><dl><dd><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 {\\egin{pmatrix}1&amp;a&amp;c\\\\\\\\0&amp;1&amp;b\\\\\\\\0&amp;0&amp;1\\\\\\\\\\\\end{pmatrix}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mo>(</mo>
<br/>            <mtable rowspacing="4pt" columnspacing="1em">
<br/>              <mtr>
<br/>                <mtd>
<br/>                  <mn>1</mn>
<br/>                </mtd>
<br/>                <mtd>
<br/>                  <mi>a</mi>
<br/>                </mtd>
<br/>                <mtd>
<br/>                  <mi>c</mi>
<br/>                </mtd>
<br/>              </mtr>
<br/>              <mtr>
<br/>                <mtd>
<br/>                  <mn>0</mn>
<br/>                </mtd>
<br/>                <mtd>
<br/>                  <mn>1</mn>
<br/>                </mtd>
<br/>                <mtd>
<br/>                  <mi>b</mi>
<br/>                </mtd>
<br/>              </mtr>
<br/>              <mtr>
<br/>                <mtd>
<br/>                  <mn>0</mn>
<br/>                </mtd>
<br/>                <mtd>
<br/>                  <mn>0</mn>
<br/>                </mtd>
<br/>                <mtd>
<br/>                  <mn>1</mn>
<br/>                </mtd>
<br/>              </mtr>
<br/>            </mtable>
<br/>            <mo>)</mo>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{pmatrix}1&amp;a&amp;c\\\\\\\\0&amp;1&amp;b\\\\\\\\0&amp;0&amp;1\\\\\\\\\\\\end{pmatrix}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f02212ea80b176e7a0290fecfa10fdbdfc36cc8b" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.005ex; width:13.018ex; height:9.176ex;" alt="\\egin{pmatrix}
<br/> 1 &amp; a &amp; c\\\\\\\\
<br/> 0 &amp; 1 &amp; b\\\\\\\\
<br/> 0 &amp; 0 &amp; 1\\\\\\\\
<br/>\\\\end{pmatrix}"></span></dd></dl></dd></dl>
<br/>under the operation of matrix multiplication. Elements <i>a, b</i> and <i>c</i> can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group").
<br/>The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to <i>n</i>-dimensional systems, and most generally, to any symplectic vector space.
<br/> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Heisenberg_group">https://en.wikipedia.org/wiki/Heisenberg_group</a>)"""@en, """En mathématiques, le <b>groupe de Heisenberg</b> d'un anneau unifère <i>A</i> (non nécessairement commutatif) est le groupe multiplicatif des matrices triangulaires supérieures de taille 3 à coefficients dans <i>A</i> et dont les éléments diagonaux sont égaux au neutre multiplicatif de l'anneau&nbsp;:
<br/>
<br/><center><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 H_{3}(A)=\\\\left\\\\{\\\\left.{\\egin{pmatrix}1&amp;a&amp;c\\\\\\\\0&amp;1&amp;b\\\\\\\\0&amp;0&amp;1\\\\\\\\\\\\end{pmatrix}}~\\
ight|~a,b,c\\\\in A\\
ight\\\\}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>H</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>A</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow>
<br/>          <mo>{</mo>
<br/>          <mrow>
<br/>            <mrow>
<br/>              <mo fence="true" stretchy="true" symmetric="true"></mo>
<br/>              <mrow>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mrow>
<br/>                    <mo>(</mo>
<br/>                    <mtable rowspacing="4pt" columnspacing="1em">
<br/>                      <mtr>
<br/>                        <mtd>
<br/>                          <mn>1</mn>
<br/>                        </mtd>
<br/>                        <mtd>
<br/>                          <mi>a</mi>
<br/>                        </mtd>
<br/>                        <mtd>
<br/>                          <mi>c</mi>
<br/>                        </mtd>
<br/>                      </mtr>
<br/>                      <mtr>
<br/>                        <mtd>
<br/>                          <mn>0</mn>
<br/>                        </mtd>
<br/>                        <mtd>
<br/>                          <mn>1</mn>
<br/>                        </mtd>
<br/>                        <mtd>
<br/>                          <mi>b</mi>
<br/>                        </mtd>
<br/>                      </mtr>
<br/>                      <mtr>
<br/>                        <mtd>
<br/>                          <mn>0</mn>
<br/>                        </mtd>
<br/>                        <mtd>
<br/>                          <mn>0</mn>
<br/>                        </mtd>
<br/>                        <mtd>
<br/>                          <mn>1</mn>
<br/>                        </mtd>
<br/>                      </mtr>
<br/>                    </mtable>
<br/>                    <mo>)</mo>
<br/>                  </mrow>
<br/>                </mrow>
<br/>                <mtext>&nbsp;</mtext>
<br/>              </mrow>
<br/>              <mo>|</mo>
<br/>            </mrow>
<br/>            <mtext>&nbsp;</mtext>
<br/>            <mi>a</mi>
<br/>            <mo>,</mo>
<br/>            <mi>b</mi>
<br/>            <mo>,</mo>
<br/>            <mi>c</mi>
<br/>            <mo>∈<!-- ∈ --></mo>
<br/>            <mi>A</mi>
<br/>          </mrow>
<br/>          <mo>}</mo>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle H_{3}(A)=\\\\left\\\\{\\\\left.{\\egin{pmatrix}1&amp;a&amp;c\\\\\\\\0&amp;1&amp;b\\\\\\\\0&amp;0&amp;1\\\\\\\\\\\\end{pmatrix}}~\\
ight|~a,b,c\\\\in A\\
ight\\\\}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c24a75e29ff1df240b194a6a18504c1b4c14b85d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.005ex; width:39.901ex; height:9.176ex;" alt="{\\\\displaystyle H_{3}(A)=\\\\left\\\\{\\\\left.{\\egin{pmatrix}1&amp;a&amp;c\\\\\\\\0&amp;1&amp;b\\\\\\\\0&amp;0&amp;1\\\\\\\\\\\\end{pmatrix}}~\\
ight|~a,b,c\\\\in A\\
ight\\\\}.}"></span></center>
<br/>Originellement, l'anneau <i>A</i> choisi par Werner Heisenberg était le corps ℝ des réels. Le «&nbsp;groupe de Heisenberg continu&nbsp;», <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 H_{3}(\\\\mathbb {R} )}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>H</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">R</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle H_{3}(\\\\mathbb {R} )}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94613d15e23941ac0013c7aa92c8d4f9b9a2f0b1" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.473ex; height:2.843ex;" alt="{\\\\displaystyle H_{3}(\\\\mathbb {R} )}"></span>, lui a permis d'expliquer, en mécanique quantique, l'équivalence entre la représentation de Heisenberg et celle de Schrödinger. On peut généraliser sa définition en géométrie symplectique.
<br/>Le «&nbsp;groupe de Heisenberg discret&nbsp;» <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 H_{3}(\\\\mathbb {Z} )}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>H</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">Z</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle H_{3}(\\\\mathbb {Z} )}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2850090a9aa975fbde4cf207decd8d26ae7b52e5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.345ex; height:2.843ex;" alt="{\\\\displaystyle H_{3}(\\\\mathbb {Z} )}"></span> correspond à l'anneau ℤ des entiers.
<br/>Le groupe de Heisenberg <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 H_{3}({\\
m {F}}_{p})}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msub>
<br/>          <mi>H</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>3</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">(</mo>
<br/>        <msub>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mrow class="MJX-TeXAtom-ORD">
<br/>              <mi mathvariant="normal">F</mi>
<br/>            </mrow>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>p</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle H_{3}({\\
m {F}}_{p})}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36af57fcdf14dc3ca09a4144c1e5970693b365d6" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:7.372ex; height:3.009ex;" alt="{\\\\displaystyle H_{3}({\\
m {F}}_{p})}"></span>, où <i>p</i> est un nombre premier, correspond au corps premier fini <b>F</b><sub><i>p</i></sub> = ℤ/<i>p</i>ℤ. C'est un <i>p</i>-groupe fini, d'ordre <i>p</i><sup>3</sup>.
<br/> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Groupe_de_Heisenberg">https://fr.wikipedia.org/wiki/Groupe_de_Heisenberg</a>)"""@fr ;
  skos:broader psr:-RMQ1RP9W-P, psr:-QDSZ76FR-B, psr:-P43HJWNV-X ;
  a skos:Concept ;
  skos:inScheme psr: ;
  dc:created "2023-08-31"^^xsd:date ;
  dc:modified "2023-08-31"^^xsd:date .