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

psr: a skos:ConceptScheme .
psr:-F1B5QL5S-0
  skos:prefLabel "algèbre non associative"@fr, "non-associative algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-PGB1XCV4-R .

psr:-PGB1XCV4-R
  skos:definition """En mathématiques, une <b>algèbre de Leibniz (droite)</b>, ainsi nommée d'après Gottfried Wilhelm Leibniz, et parfois appelée <b>algèbre de Loday</b>, d'après Jean-Louis Loday, est un module <i>L</i> sur un anneau commutatif <i>R</i> muni d'un produit bilinéaire [-,-], appelé crochet, satisfaisant <b>l'identité de Leibniz</b>
         
         <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 [[a,b],c]=[a,[b,c]]+[[a,c],b].\\\\,}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">[</mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">]</mo>
         <mo>=</mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mo stretchy="false">[</mo>
         <mi>b</mi>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">]</mo>
         <mo stretchy="false">]</mo>
         <mo>+</mo>
         <mo stretchy="false">[</mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">]</mo>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         <mo>.</mo>
         <mspace width="thinmathspace"></mspace>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle [[a,b],c]=[a,[b,c]]+[[a,c],b].\\\\,}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd08048239d488970159d6c46f1126044ba63cd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.641ex; height:2.843ex;" alt="{\\\\displaystyle [[a,b],c]=[a,[b,c]]+[[a,c],b].\\\\,}"></span></dd></dl>
         En d'autres termes, la multiplication à droite par un élément <i>c</i> est une dérivation. Si, de plus, le crochet est alterné (i.e. [<i>a</i>, <i>a</i>] = 0) alors l'algèbre de Leibniz est une algèbre de Lie. En effet, dans ce cas [<i>a</i>, <i>b</i>] = −[<i>b</i>, <i>a</i>] et l'identité de Leibniz est équivalente à l'identité de Jacobi ([<i>a</i>, [<i>b</i>, <i>c</i>]] + [<i>c</i>, [<i>a</i>, <i>b</i>]] + [<i>b</i>, [<i>c</i>, <i>a</i>]] = 0).
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Leibniz">https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Leibniz</a>)"""@fr, """In mathematics, a <b>(right) Leibniz algebra</b>, named after Gottfried Wilhelm Leibniz, sometimes called a <b>Loday algebra</b>, after Jean-Louis Loday, is a module <i>L</i> over a commutative ring  <i>R</i> with a bilinear product [ _ , _ ] satisfying the <b>Leibniz identity</b>
         
         <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 [[a,b],c]=[a,[b,c]]+[[a,c],b].\\\\,}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">[</mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">]</mo>
         <mo>=</mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mo stretchy="false">[</mo>
         <mi>b</mi>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">]</mo>
         <mo stretchy="false">]</mo>
         <mo>+</mo>
         <mo stretchy="false">[</mo>
         <mo stretchy="false">[</mo>
         <mi>a</mi>
         <mo>,</mo>
         <mi>c</mi>
         <mo stretchy="false">]</mo>
         <mo>,</mo>
         <mi>b</mi>
         <mo stretchy="false">]</mo>
         <mo>.</mo>
         <mspace width="thinmathspace"></mspace>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle [[a,b],c]=[a,[b,c]]+[[a,c],b].\\\\,}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd08048239d488970159d6c46f1126044ba63cd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.641ex; height:2.843ex;" alt="[[a,b],c]=[a,[b,c]]+[[a,c],b].\\\\,"></span></dd></dl>
         In other words, right multiplication by any element <i>c</i> is a derivation. If in addition the bracket is alternating ([<i>a</i>, <i>a</i>] = 0) then the Leibniz algebra is a Lie algebra. Indeed, in this case [<i>a</i>, <i>b</i>] = −[<i>b</i>, <i>a</i>] and the Leibniz's identity is equivalent to Jacobi's identity ([<i>a</i>, [<i>b</i>, <i>c</i>]] + [<i>c</i>, [<i>a</i>, <i>b</i>]] + [<i>b</i>, [<i>c</i>, <i>a</i>]] = 0). Conversely any Lie algebra is obviously a Leibniz algebra.
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Leibniz_algebra">https://en.wikipedia.org/wiki/Leibniz_algebra</a>)"""@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Leibniz_algebra>, <https://fr.wikipedia.org/wiki/Alg%C3%A8bre_de_Leibniz> ;
  skos:altLabel "algèbre de Loday"@fr, "Loday algebra"@en ;
  skos:inScheme psr: ;
  skos:broader psr:-FBT35M65-C, psr:-F1B5QL5S-0 ;
  dc:modified "2023-07-26"^^xsd:date ;
  skos:prefLabel "algèbre de Leibniz"@fr, "Leibniz algebra"@en ;
  dc:created "2023-07-26"^^xsd:date ;
  a skos:Concept .