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href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="lie_algebras">Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-algebras</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#semistrict_case'>Semistrict case</a></li> <li><a href='#strict_case'>Strict case</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A Lie 2-algebra is to a <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a> as a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> is to a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. Thus, it is a <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>.</p> <h2 id="definition">Definition</h2> <h3 id="semistrict_case">Semistrict case</h3> <p>A (“semistrict”) Lie 2-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a> with generators concentrated in the lowest two degrees.</p> <p>This means that it is</p> <ul> <li> <p>a pair of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>𝔤</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_0, \mathfrak{g}_1</annotation></semantics></math></p> </li> <li> <p>equipped with <a class="existingWikiWord" href="/nlab/show/linear+functions">linear functions</a> as follows:</p> <p>a unary bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-]</annotation></semantics></math> encoding a <em><a class="existingWikiWord" href="/nlab/show/differential">differential</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>:</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \delta : \mathfrak{g}_1 \to \mathfrak{g}_0 \, </annotation></semantics></math></div> <p>and a binary bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math>, whose component on elements in degree 0 is a <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>∨</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> [-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_0 </annotation></semantics></math></div> <p>and whose component on elements in degree 0 and degree 1 is a <em>weak <a class="existingWikiWord" href="/nlab/show/action">action</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>⊗</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> \alpha(-,-) : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to \mathfrak{g}_1 \,; </annotation></semantics></math></div> <p>and a trinary bracket</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>∨</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>∨</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> [-,-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_1 </annotation></semantics></math></div> <p>called the <em>Jacobiator</em>;</p> </li> <li> <p>such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-,-]</annotation></semantics></math> are skew-symmetric in their arguments, as indicated;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> respects the brackets: for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>𝔤</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x \in \mathfrak{g}_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><msub><mi>𝔤</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">h \in \mathfrak{g}_1</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>δ</mi><mi>h</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \delta [x,h] = [x, \delta h] </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>δ</mi><mi>h</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> \delta \alpha(x,h) = [x, \delta h] \,; </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> holds up to the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi></mrow><annotation encoding="application/x-tex">\delta</annotation></semantics></math> of the <em>Jacobiator</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-,-]</annotation></semantics></math>: for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><msub><mi>𝔤</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">x,y,z \in \mathfrak{g}_0</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>z</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mi>δ</mi><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = \delta [x,y,z] </annotation></semantics></math></div></li> <li> <p>as does the <a class="existingWikiWord" href="/nlab/show/action">action</a> property:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>−</mo><mi>α</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>δ</mi><mi>h</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \alpha(x,[y,h]) - \alpha(y,[x,h]) = \alpha([x,y],h) + [x,y,\delta h] </annotation></semantics></math></div></li> <li> <p>the Jacobiator is <a class="existingWikiWord" href="/nlab/show/coherence+law">coherent</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>w</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>w</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>w</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>w</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>,</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [[w,x,y], z] + [[w,y,z],x] + [[w,y],x,z] + [[x,z],w,y] = [[w,x,z], y] + [[x,y,z], w] + [[w,x],y,z] + [[w,z], x,y] + [[x,y], w,z] + [[y,z],w,x] \,. </annotation></semantics></math></div></li> </ul> </li> </ul> <p>The Jacobiator identity equivalently expresses the <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutativity</a> of the following <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in the given <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a> (analogous to the <a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon identity</a>)</p> <p><img src="https://ncatlab.org/nlab/files/JacobiatorIdentity.jpg" width="560" /></p> <blockquote> <p>(graphics grabbed from <a href="#BaezCrans04">Baez-Crans 04, p. 19</a>)</p> </blockquote> <h3 id="strict_case">Strict case</h3> <p>If the trinary bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-,-]</annotation></semantics></math> in a Lie 2-algebra is trivial, one speaks of a <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a>. Strict Lie 2-algebras are equivalently <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a>s (see there for details).</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation+Lie+2-algebra">derivation Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4d+supergravity+Lie+2-algebra">4d supergravity Lie 2-algebra</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><strong>Lie 2-algebra</strong>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/membrane+matrix+model">membrane matrix model</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+3-group">Lie 3-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid">L-∞ algebroid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="BaezCrans04"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Alissa+Crans">Alissa Crans</a>, <em>Higher-Dimensional Algebra VI: Lie 2-Algebras</em> Theory and Applications of Categories, Vol. 12, (2004) No. 15, pp 492-528. (<a href="http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html">TAC:12-15</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <a class="existingWikiWord" href="/nlab/show/Eugene+Lerman">Eugene Lerman</a>, <em>Lie 2-algebras of vector fields</em>, <a href="http://arxiv.org/abs/1609.03944">arxiv/1609.03944</a></p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/weak+Lie+2-algebras">weak Lie 2-algebras</a>:</p> <ul> <li id="Roytenberg07"><a class="existingWikiWord" href="/nlab/show/Dmitry+Roytenberg">Dmitry Roytenberg</a>, <em>On weak Lie 2-algebras</em>, AIP Conference Proceedings 956, 180 (2007) (<a href="http://arxiv.org/abs/0712.3461">arXiv:0712.3461</a>, <a href="https://doi.org/10.1063/1.2820967">doi:10.1063/1.2820967</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 17, 2022 at 16:07:28. 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