@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:-L16ZWD33-J
  skos:prefLabel "Bianchi classification"@en, "classification de Bianchi"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr: a skos:ConceptScheme .
psr:-PR2WWWLH-4
  skos:prefLabel "algèbre de Lie graduée"@fr, "graded Lie algebra"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-XS63QZ07-V
  skos:prefLabel "Kostant's convexity theorem"@en, "théorème de la convexité de Kostant"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-KSGTQ4H9-8
  skos:prefLabel "Cartan decomposition"@en, "décomposition de Cartan"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-JTHL7QX2-Q
  skos:prefLabel "Casimir operator"@en, "opérateur de Casimir"@fr ;
  a skos:Concept ;
  skos:related psr:-FBT35M65-C .

psr:-NBSLMP4T-3
  skos:prefLabel "cône convexe invariant"@fr, "invariant convex cone"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-D9KT6PSR-Z
  skos:prefLabel "Lie bracket"@en, "crochet de Lie"@fr ;
  a skos:Concept ;
  skos:related psr:-FBT35M65-C .

psr:-PZB19WL5-6
  skos:prefLabel "Lie's third theorem"@en, "troisième théorème de Lie"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-GHJDNWMW-1
  skos:prefLabel "Vogel plane"@en, "plan de Vogel"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-NQBFGBJ1-6
  skos:prefLabel "root system"@en, "système de racines"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-RXQC777M-K
  skos:prefLabel "algèbre différentielle"@fr, "differential algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-FBT35M65-C .

psr:-P64TCLGF-0
  skos:prefLabel "groupe de Weyl"@fr, "Weyl group"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-K4DNK8ST-D
  skos:prefLabel "contraction de groupe"@fr, "group contraction"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-DNZR7BH2-W
  skos:prefLabel "algèbre de Virasoro"@fr, "Virasoro algebra"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-RMQ1RP9W-P
  skos:prefLabel "groupe de Lie"@fr, "Lie group"@en ;
  a skos:Concept ;
  skos:related psr:-FBT35M65-C .

psr:-G1GVN6FR-3
  skos:prefLabel "Kantor-Koecher-Tits construction"@en, "construction de Kantor-Koecher-Tits"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-R553W8RM-J
  skos:prefLabel "vertex operator algebra"@en, "algèbre vertex"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-CS2FD3K2-0
  skos:prefLabel "algebra over a field"@en, "algèbre sur un corps"@fr ;
  a skos:Concept ;
  skos:narrower psr:-FBT35M65-C .

psr:-KFJZXBV9-1
  skos:prefLabel "algèbre de Lie d'homotopie"@fr, "homotopy Lie algebra"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-FBT35M65-C
  skos:inScheme psr: ;
  skos:narrower psr:-PZB19WL5-6, psr:-GMQS050D-B, psr:-NQBFGBJ1-6, psr:-B8WV4BTX-0, psr:-CCFD3TTQ-6, psr:-G1GVN6FR-3, psr:-KRMCJZW3-1, psr:-P64TCLGF-0, psr:-KFJZXBV9-1, psr:-PGB1XCV4-R, psr:-GHJDNWMW-1, psr:-GT0RZ5JJ-1, psr:-XS63QZ07-V, psr:-L16ZWD33-J, psr:-R553W8RM-J, psr:-KSGTQ4H9-8, psr:-K4DNK8ST-D, psr:-GT54GL92-Q, psr:-NBSLMP4T-3, psr:-PR2WWWLH-4, psr:-DNZR7BH2-W ;
  a skos:Concept ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Lie>, <https://en.wikipedia.org/wiki/Lie_algebra> ;
  skos:related psr:-RMQ1RP9W-P, psr:-D9KT6PSR-Z, psr:-JTHL7QX2-Q ;
  skos:definition """En mathématiques, une algèbre de Lie, nommée en l'honneur du mathématicien Sophus Lie, est un espace vectoriel qui est muni d'un crochet de Lie, c'est-à-dire d'une loi de composition interne bilinéaire, alternée, et qui vérifie la relation de Jacobi. Une algèbre de Lie est un cas particulier d'algèbre sur un corps. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Lie">https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Lie</a>)"""@fr, """In mathematics, a <b>Lie algebra</b> (pronounced <span class="rt-commentedText nowrap"><span class="IPA nopopups noexcerpt" lang="en-fonipa"><a href="/wiki/Help:IPA/English" title="Help:IPA/English">/<span style="border-bottom:1px dotted"><span title="'l' in 'lie'">l</span><span title="/iː/: 'ee' in 'fleece'">iː</span></span>/</a></span></span> <i title="English pronunciation respelling"><span style="font-size:90%">LEE</span></i>) is a 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 {\\\\mathfrak {g}}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="fraktur">g</mi>
         </mrow>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {g}}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\\\\mathfrak {g}}"></span> together with an operation called the <b>Lie bracket</b>, an alternating bilinear map <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 {\\\\mathfrak {g}}\\	imes {\\\\mathfrak {g}}\\
ightarrow {\\\\mathfrak {g}}}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mrow class="MJX-TeXAtom-ORD">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="fraktur">g</mi>
         </mrow>
         </mrow>
         <mo>×<!-- × --></mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="fraktur">g</mi>
         </mrow>
         </mrow>
         <mo stretchy="false">→<!-- → --></mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="fraktur">g</mi>
         </mrow>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle {\\\\mathfrak {g}}\\	imes {\\\\mathfrak {g}}\\
ightarrow {\\\\mathfrak {g}}}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43a4432b80e75d9303120e999ddd0fca588156f8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.97ex; height:2.176ex;" alt="{\\\\displaystyle {\\\\mathfrak {g}}\\	imes {\\\\mathfrak {g}}\\
ightarrow {\\\\mathfrak {g}}}"></span>, that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors <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}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>x</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle x}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x"></span> and <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 y}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>y</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle y}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="y"></span> is denoted <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,y]}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">[</mo>
         <mi>x</mi>
         <mo>,</mo>
         <mi>y</mi>
         <mo stretchy="false">]</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle [x,y]}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7bd6292c6023626c6358bfd3943a031b27d663" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.813ex; height:2.843ex;" alt="[x,y]"></span>. The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative.
         Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining <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,y]=xy-yx}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">[</mo>
         <mi>x</mi>
         <mo>,</mo>
         <mi>y</mi>
         <mo stretchy="false">]</mo>
         <mo>=</mo>
         <mi>x</mi>
         <mi>y</mi>
         <mo>−<!-- − --></mo>
         <mi>y</mi>
         <mi>x</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle [x,y]=xy-yx}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42b4220c8122ebd2a21c517ca80639581679cfa6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.722ex; height:2.843ex;" alt="{\\\\displaystyle [x,y]=xy-yx}"></span> correctly defines a Lie bracket in addition to the already existing multiplication operation.
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Lie_algebra">https://en.wikipedia.org/wiki/Lie_algebra</a>)"""@en ;
  skos:prefLabel "Lie algebra"@en, "algèbre de Lie"@fr ;
  skos:broader psr:-CS2FD3K2-0, psr:-RXQC777M-K ;
  dc:modified "2023-08-23"^^xsd:date .

psr:-KRMCJZW3-1
  skos:prefLabel "symmetric cone"@en, "cône symétrique"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-GT0RZ5JJ-1
  skos:prefLabel "differential graded Lie algebra"@en, "algèbre différentielle de Lie graduée"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-GT54GL92-Q
  skos:prefLabel "Killing form"@en, "forme de Killing"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-B8WV4BTX-0
  skos:prefLabel "algèbre de Lie spéciale linéaire"@fr, "special linear Lie algebra"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-GMQS050D-B
  skos:prefLabel "forme réelle"@fr, "real form"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-PGB1XCV4-R
  skos:prefLabel "Leibniz algebra"@en, "algèbre de Leibniz"@fr ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .

psr:-CCFD3TTQ-6
  skos:prefLabel "algèbre de Kac-Moody"@fr, "Kac-Moody algebra"@en ;
  a skos:Concept ;
  skos:broader psr:-FBT35M65-C .