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

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

psr: a skos:ConceptScheme .
psr:-H8HFSDFX-5
  skos:prefLabel "loi de composition interne"@fr, "internal binary operation"@en ;
  a skos:Concept ;
  skos:narrower psr:-D9KT6PSR-Z .

psr:-D9KT6PSR-Z
  skos:exactMatch <https://en.wikipedia.org/wiki/Lie_algebra>, <https://fr.wikipedia.org/wiki/Crochet_de_Lie> ;
  a skos:Concept ;
  skos:broader psr:-H8HFSDFX-5 ;
  skos:prefLabel "crochet de Lie"@fr, "Lie bracket"@en ;
  skos:definition """In mathematics, a <b>Lie algebra</b> (pronounced /liː/ <i>LEE</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, """Un crochet de Lie est une loi de composition interne [∙, ∙] sur un espace vectoriel, qui lui confère une structure d'algèbre de Lie. Le commutateur de deux endomorphismes <i>u</i> et <i>v</i>, noté [<i>u</i>, <i>v</i>] = <i>u</i><i>v</i> – <i>v</i><i>u</i>, est l'un des exemples les plus simples. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Crochet_de_Lie">https://fr.wikipedia.org/wiki/Crochet_de_Lie</a>)"""@fr ;
  skos:related psr:-FBT35M65-C ;
  skos:inScheme psr: .