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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #37 to #38: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='lie_theory'><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Lie theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+theory'>∞-Lie theory</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/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/diff/generalized+smooth+space'>generalized smooth space</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher space</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/concrete+smooth+infinity-groupoid'>concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-groupoid'>2-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+theory'>Lie theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+integration'>Lie integration</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+differentiation'>Lie differentiation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie%27s+three+theorems'>Lie's three theorems</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Lie+n-groupoid'>∞-Lie groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+n-groupoid'>strict ∞-Lie groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+groupoid'>Lie groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+stack'>differentiable stack</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orbifold'>orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+group'>∞-Lie group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/simple+Lie+group'>simple Lie group</a>, <a class='existingWikiWord' href='/nlab/show/diff/semisimple+Lie+group'>semisimple Lie group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Lie+infinity-algebroid'>∞-Lie algebroid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebroid'>Lie algebroid</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+infinity-algebroid+representation'>Lie ∞-algebroid representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞-algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+L-infinity+algebras'>model structure for L-∞ algebras</a>: <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-Lie+algebras'>on dg-Lie algebras</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-coalgebras'>on dg-coalgebras</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+Lie+algebras'>on simplicial Lie algebras</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/semisimple+Lie+algebra'>semisimple Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+algebra'>compact Lie algebra</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+2-algebra'>Lie 2-algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/strict+Lie+2-algebra'>strict Lie 2-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+crossed+module'>differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+3-algebra'>Lie 3-algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+2-crossed+module'>differential 2-crossed module</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+graded+Lie+algebra'>dg-Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+Lie+algebra'>simplicial Lie algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+L-infinity+algebra'>super L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/formal+group'>formal group</a>, <a class='existingWikiWord' href='/nlab/show/diff/formal+groupoid'>formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chevalley-Eilenberg+algebra'>Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Weil+algebra'>Weil algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/invariant+polynomial'>invariant polynomial</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Killing+form'>Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_2' xmlns='http://www.w3.org/1998/Math/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/diff/Atiyah+Lie+groupoid'>Atiyah Lie groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+n-groupoid'>path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_3' xmlns='http://www.w3.org/1998/Math/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/diff/orthogonal+group'>orthogonal group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spin+group'>spin group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+n-groupoid'>circle Lie n-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/circle+group'>circle group</a></li> </ul> </li> </ul> <p><em><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_4' xmlns='http://www.w3.org/1998/Math/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/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action+Lie+algebroid'>action Lie algebroid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Atiyah+Lie+algebroid'>Atiyah Lie algebroid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+Lie+n-algebroid'>symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+manifold'>symplectic manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Poisson+Lie+algebroid'>Poisson Lie algebroid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Courant+algebroid'>Courant Lie algebroid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/generalized+complex+geometry'>generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_5' xmlns='http://www.w3.org/1998/Math/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/diff/general+linear+Lie+algebra'>general linear Lie algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+Lie+algebra'>orthogonal Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+Lie+algebra'>special orthogonal Lie algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/endomorphism+dg-Lie+algebra'>endomorphism L-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-Lie+algebra'>automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/String+Lie+2-algebra'>string Lie 2-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fivebrane+Lie+6-algebra'>fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/D%3D11+supergravity+Lie+3-algebra'>supergravity Lie 3-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/supergravity+Lie+6-algebra'>supergravity Lie 6-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+Lie-n+algebra'>line Lie n-algebra</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#GradingConvention'>Grading conventions</a></li><li><a href='#OfLieAlgebra'>Of Lie algebras</a><ul><li><a href='#DefForLieAlg'>Definition</a></li><li><a href='#PropertiesForLieAlg'>Properties</a></li></ul></li><li><a href='#of_algebras'>Of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebras</a></li><li><a href='#OfLieAlgebroids'>Of Lie algebroids</a></li><li><a href='#of_lie_algebroids_2'>Of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Lie algebroids</a></li><li><a href='#examples'>Examples</a><ul><li><a href='#of_abelian_lie_algebras'>Of abelian Lie <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-algebras</a></li><li><a href='#of_'>Of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathfrak{su}(2)</annotation></semantics></math></a></li><li><a href='#of_the_tangent_lie_algebroid_'>Of the tangent Lie algebroid <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math></a></li><li><a href='#of_the_string_lie_2algebra'>Of the string Lie 2-algebra</a></li><li><a href='#weil_algebra'>Weil algebra</a></li><li><a href='#lie_algebra_cohomology'>Lie algebra cohomology</a></li><li><a href='#brst_complex'>BRST complex</a></li></ul></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>The <em>Chevalley-Eilenberg algebra</em> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CE(\mathfrak{g})</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a> is a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>differential graded algebra</a> of elements dual to <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a> encodes the Lie bracket on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/chain+homology+and+cohomology'>cochain cohomology</a> of the underlying <a class='existingWikiWord' href='/nlab/show/diff/cochain+complex'>cochain complex</a> is the <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a> of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math>.</p> <p>This generalizes to a notion of Chevalley-Eilenberg algebra for <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞-algebra</a>, a <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebroid'>Lie algebroid</a> and generally an <a class='existingWikiWord' href='/nlab/show/diff/Lie+infinity-algebroid'>∞-Lie algebroid</a>.</p> <h2 id='GradingConvention'>Grading conventions</h2> <p>This differential-graded subject is somewhat notorious for a plethora of equivalent but different conventions on gradings and signs.</p> <p>For the following we adopt the convention that for <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> an <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/graded+vector+space'>graded vector space</a> we write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi></mtd> <mtd><mo>:</mo><mo>=</mo><mi>Sym</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mi>k</mi><mo>⊕</mo><mo stretchy='false'>(</mo><msub><mi>V</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>⊕</mo><mo stretchy='false'>(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>V</mi> <mn>0</mn></msub><mo>∧</mo><msub><mi>V</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>⊕</mo><mo stretchy='false'>(</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>V</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>V</mi> <mn>0</mn></msub><mo>∧</mo><msub><mi>V</mi> <mn>0</mn></msub><mo>∧</mo><msub><mi>V</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo><mo>⊕</mo><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} \wedge^\bullet V &:= Sym(V[1]) \\ & = k \oplus (V_0) \oplus (V_1 \oplus V_0 \wedge V_0) \oplus (V_2 \oplus V_1 \wedge V_0 \oplus V_0 \wedge V_0 \wedge V_0) \oplus \cdots \end{aligned} </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/free+functor'>free</a> graded-commutative algebra on the graded vector space obtained by shifting <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> <em>up</em> in degree by one.</p> <p>Here the elements in the <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th term in parenthesis are in degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>.</p> <p>A plain vector space, such as the dual <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math><span> of the vector space underlying a Lie algebra,<del class='diffmod'> we</del><ins class='diffmod'> is</ins><del class='diffmod'> regard</del><ins class='diffmod'> regarded</ins><del class='diffdel'> her</del> as a</span><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math><span> -graded vector space in degree 0.<del class='diffmod'> For</del><ins class='diffmod'> The</ins><del class='diffmod'> such,</del><ins class='diffmod'> quantity</ins></span><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math> is the ordinary <a class='existingWikiWord' href='/nlab/show/diff/exterior+algebra'>Grassmann algebra</a> over <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math>, where elements of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math> are generators of degree 1.</p> <h2 id='OfLieAlgebra'>Of Lie algebras</h2> <h3 id='DefForLieAlg'>Definition</h3> <p>The <em>Chevalley-Eilenberg algebra</em> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CE(\mathfrak{g})</annotation></semantics></math> of a finite dimensional <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree</a> graded-commutative <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> whose underlying graded algebra is the <a class='existingWikiWord' href='/nlab/show/diff/exterior+algebra'>Grassmann algebra</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>=</mo><mi>k</mi><mo>⊕</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>⊕</mo><mo stretchy='false'>(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo><mo>⊕</mo><mi>⋯</mi></mrow><annotation encoding='application/x-tex'> \wedge^\bullet \mathfrak{g}^* = k \oplus \mathfrak{g}^* \oplus (\mathfrak{g}^* \wedge \mathfrak{g}^* ) \oplus \cdots </annotation></semantics></math></div> <p>(with the <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>th skew-symmetrized power in degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math><span><del class='diffmod'> )</del><ins class='diffmod'> ),</ins><ins class='diffins'> and</ins><ins class='diffins'> whose</ins></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a></ins><ins class='diffins'> </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math></ins><ins class='diffins'> (of degree +1) is defined on </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math></ins><ins class='diffins'> as the dual of the Lie bracket</ins></p><span /><del class='diffmod'><p>and whose <a class='existingWikiWord' href='/nlab/show/diff/differential'>differential</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> (of degree +1) is on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math> the dual of the Lie bracket</p></del><ins class='diffmod'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><msub><mo stretchy='false'>|</mo> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>]</mo> <mo>*</mo></msup><mo>:</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>,</mo></mrow><annotation encoding='application/x-tex'> d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*, </annotation></semantics></math></div></ins> <del class='diffmod'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><msub><mo stretchy='false'>|</mo> <mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow></msub><mo>:</mo><mo>=</mo><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>]</mo> <mo>*</mo></msup><mo>:</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'> d|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* </annotation></semantics></math></div></del><ins class='diffmod'><p>and extended uniquely as a graded <a class='existingWikiWord' href='/nlab/show/diff/derivation'>derivation</a> on <math class='maruku-mathml' display='inline' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math>.</p></ins><span /><p><span><del class='diffmod'> extended</del><ins class='diffmod'> That</ins><del class='diffmod'> uniquely</del><ins class='diffmod'> this</ins><del class='diffmod'> as</del><ins class='diffmod'> differential</ins><del class='diffmod'> a</del><ins class='diffmod'> indeed</ins><del class='diffmod'> graded</del><ins class='diffmod'> squares</ins><ins class='diffins'> to</ins><ins class='diffins'> 0,</ins></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/derivation'>derivation</a></del><ins class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∘</mo><mi>d</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>d \circ d = 0</annotation></semantics></math></ins><span><del class='diffmod'> </del><ins class='diffmod'> ,</ins><del class='diffmod'> on</del><ins class='diffmod'> is</ins><ins class='diffins'> precisely</ins><ins class='diffins'> the</ins><ins class='diffins'> fact</ins><ins class='diffins'> that</ins><ins class='diffins'> the</ins><ins class='diffins'> Lie</ins><ins class='diffins'> bracket</ins><ins class='diffins'> satisfies</ins><ins class='diffins'> the</ins></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Jacobi+identity'>Jacobi identity</a></ins>.</p> <p><span><del class='diffmod'> That</del><ins class='diffmod'> If</ins><del class='diffmod'> this</del><ins class='diffmod'> we</ins><del class='diffmod'> differential</del><ins class='diffmod'> choose</ins><del class='diffmod'> indeed</del><ins class='diffmod'> a</ins><del class='diffmod'> squares</del><ins class='diffmod'> dual</ins><del class='diffmod'> to</del><ins class='diffmod'> basis</ins><del class='diffdel'> 0,</del></span><math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><del class='diffmod'><mi>d</mi></del><ins class='diffmod'><mo stretchy='false'>{</mo></ins><del class='diffmod'><mo>∘</mo></del><ins class='diffmod'><msup><mi>t</mi> <mi>a</mi></msup></ins><del class='diffmod'><mi>d</mi></del><ins class='diffmod'><mo stretchy='false'>}</mo></ins><del class='diffdel'><mo>=</mo></del><del class='diffdel'><mn>0</mn></del></mrow><annotation encoding='application/x-tex'><span><del class='diffmod'> d</del><ins class='diffmod'> \{t^a\}</ins><del class='diffdel'> \circ</del><del class='diffdel'> d</del><del class='diffdel'> =</del><del class='diffdel'> 0</del></span></annotation></semantics></math><span><del class='diffmod'> ,</del><ins class='diffmod'> </ins><del class='diffmod'> is</del><ins class='diffmod'> of</ins><del class='diffdel'> precisely</del><del class='diffdel'> the</del><del class='diffdel'> fact</del><del class='diffdel'> that</del><del class='diffdel'> the</del><del class='diffdel'> Lie</del><del class='diffdel'> bracket</del><del class='diffdel'> satisfies</del><del class='diffdel'> the</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Jacobi+identity'>Jacobi identity</a></del><ins class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math></ins><span><del class='diffmod'> .</del><ins class='diffmod'> </ins><ins class='diffins'> and</ins><ins class='diffins'> let</ins></span><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow /> <mrow><mi>b</mi><mi>c</mi></mrow></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{C^a{}_{b c}\}</annotation></semantics></math></ins><ins class='diffins'> be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is</ins></p><span /><del class='diffmod'><p>If we choose a dual basis <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msup><mi>t</mi> <mi>a</mi></msup><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{t^a\}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math> and let <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow /> <mrow><mi>b</mi><mi>c</mi></mrow></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{C^a{}_{b c}\}</annotation></semantics></math> be the structure constants of the Lie bracket in that basis, then the action of the differential on the basis generators is</p></del><ins class='diffmod'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_730bc7ed8dfe58eff7327943a16fdeec2a8edb0c_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><msup><mi>t</mi> <mi>a</mi></msup><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow /> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>t</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>t</mi> <mi>c</mi></msup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,, </annotation></semantics></math></div></ins> <del class='diffdel'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><msup><mi>t</mi> <mi>a</mi></msup><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>C</mi> <mi>a</mi></msup><msub><mrow /> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msup><mi>t</mi> <mi>b</mi></msup><mo>∧</mo><msup><mi>t</mi> <mi>c</mi></msup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> d t^a = - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c \,, </annotation></semantics></math></div></del><del class='diffdel'> </del><p>where here and in the following a sum over repeated indices is implicit.</p> <p>This has a more or less evident generalization to infinite-dimensional Lie algebras.,</p> <h3 id='PropertiesForLieAlg'>Properties</h3> <p>One observes that for <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> a vector space, the graded-commutative dg-algebra structures on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math> are precisely in bijection with Lie algebra structures on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math>: the dual of the restriction of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math> defines a skew-linear bracket and the condition <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>d^2 = 0</annotation></semantics></math> holds if and only if that bracket satisfies the Jacobi identity.</p> <p>Moreover, morphisms if Lie algebras <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi><mo>→</mo><mi>𝔥</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g} \to \mathfrak{h}</annotation></semantics></math> are precisely in bijection with morphisms of dg-algebras <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>←</mo><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔥</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CE(\mathfrak{g}) \leftarrow CE(\mathfrak{h})</annotation></semantics></math>. And the <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi></mrow><annotation encoding='application/x-tex'>CE</annotation></semantics></math>-construction is functorial.</p> <p>Therefore, if we write <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>dgAlg</mi> <mrow><mi>sf</mi><mo>,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>dgAlg_{sf,1}</annotation></semantics></math> for the category whose objects are <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dga</a>s on generators in degree 1, we find that the construction of CE-algebras from Lie algebras constitutes a canonical <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence of categories</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>LieAlg</mi><mover><munder><mo>→</mo><mo>≃</mo></munder><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></mover><mo stretchy='false'>(</mo><msub><mi>dgAlg</mi> <mrow><mi>sf</mi><mo>,</mo><mn>1</mn></mrow></msub><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> LieAlg \stackrel{CE(-)}{\underset{\simeq}{\to}} (dgAlg_{sf,1})^{op} \,, </annotation></semantics></math></div> <p>where on the right we have the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a>.</p> <p>This says that in a sense the Chevalley-Eilenberg algebra is just another way of looking at (finite dimensional) Lie algebras.</p> <p>There is an analogous statement not involving the dualization: Lie algebra structures on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> are also in bijection with the structure of a <em><a class='existingWikiWord' href='/nlab/show/diff/differential+graded+coalgebra'>differential graded coalgebra</a></em> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\vee^\bullet \mathfrak{g}, D)</annotation></semantics></math> on the free graded-co-commutative coalgebra <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\vee^\bullet \mathfrak{g}</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/derivation'>derivation</a> of degree -1 squaring to 0.</p> <p>The relation between the differentials is simply dualization</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi><mo>,</mo><mi>D</mi><mo stretchy='false'>)</mo><mo>↔</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (\vee^\bullet \mathfrak{g}, D) \leftrightarrow (\wedge^\bullet \mathfrak{g}^* , d ) </annotation></semantics></math></div> <p>where for each <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\omega \in \wedge^\bullet \mathfrak{g}^*</annotation></semantics></math> we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mi>ω</mi><mo>=</mo><mi>ω</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> d \omega = \omega(D(-)) \,. </annotation></semantics></math></div> <h2 id='of_algebras'>Of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebras</h2> <p>The equivalence between Lie algebras and differential graded algebras/coalgebras discussed above suggests a grand generalization by simply generalizing the <a class='existingWikiWord' href='/nlab/show/diff/exterior+algebra'>Grassmann algebra</a> over a <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math> to the Grassmann algebra over a <a class='existingWikiWord' href='/nlab/show/diff/graded+vector+space'>graded vector space</a>.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> is a graded vector space, then a differential <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> of degree -1 squaring to 0 on <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\vee^\bullet \mathfrak{g}</annotation></semantics></math> is precisely the same as equipping <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> with the structure of an <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a>.</p> <p>Dually, this corresponds to a general <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dga</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>,</mo><mi>d</mi><mo>=</mo><msup><mi>D</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d = D^*) \,. </annotation></semantics></math></div> <p>This we may usefully think of as the Chevalley-Eilenberg algebra of the <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebra <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math>.</p> <p>So <em>every</em> commutative <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dga</a> (degreewise finite-dimensional) is the Chevaley-Eilenberg algebra of some <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a> of finite type.</p> <p>This means that many constructions involving <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a>s are secretly about <a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+theory'>∞-Lie theory</a>. For instance the <a class='existingWikiWord' href='/nlab/show/diff/Sullivan+construction'>Sullivan construction</a> in <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a> may be interpreted in terms of <a class='existingWikiWord' href='/nlab/show/diff/Lie+integration'>Lie integration</a> of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebras.</p> <h2 id='OfLieAlgebroids'>Of Lie algebroids</h2> <p>For <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔞</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{a}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebroid'>Lie algebroid</a> given as</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E\to X</annotation></semantics></math></p> </li> <li> <p>with anchor map <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>\rho : E \to T X</annotation></semantics></math></p> </li> <li> <p>and bracket <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_73' xmlns='http://www.w3.org/1998/Math/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><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Γ</mi><mo stretchy='false'>(</mo><mi>E</mi><mo stretchy='false'>)</mo><msub><mo>∧</mo> <mi>ℝ</mi></msub><mi>Γ</mi><mo stretchy='false'>(</mo><mi>E</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Γ</mi><mo stretchy='false'>(</mo><mi>E</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[-,-] \;\colon\; \Gamma(E)\wedge_{\mathbb{R}} \Gamma(E) \to \Gamma(E)</annotation></semantics></math></p> </li> </ul> <p>the corresponding Chevalley-Eilenberg algebra is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔞</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mrow><mo>(</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow> <mo>•</mo></msubsup><mi>Γ</mi><mo stretchy='false'>(</mo><mi>E</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mo>,</mo><mi>d</mi><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> CE(\mathfrak{a}) := \left(\wedge^\bullet_{C^\infty(X)} \Gamma(E)^*, d\right) \,, </annotation></semantics></math></div> <p>where now the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a>s and dualization is over the ring <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\infty(X)</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/smooth+map'>smooth function</a>s on the base space <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (with values in the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real number</a>s). The differential <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> is given by the formula</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>d</mi><mi>ω</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>e</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>e</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>Shuff</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></munder><mi>sgn</mi><mo stretchy='false'>(</mo><mi>σ</mi><mo stretchy='false'>)</mo><mi>ρ</mi><mo stretchy='false'>(</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>ω</mi><mo stretchy='false'>(</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>+</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>Shuff</mi><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></munder><mi>sign</mi><mo stretchy='false'>(</mo><mi>σ</mi><mo stretchy='false'>)</mo><mi>ω</mi><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>]</mo><mo>,</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>e</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> (d\omega)(e_0, \cdots, e_n) = \sum_{\sigma \in Shuff(1,n)} sgn(\sigma) \rho(e_{\sigma(0)})(\omega(e_{\sigma(1)}, \cdots, e_{\sigma(n)})) + \sum_{\sigma \in Shuff(2,n-1)} sign(\sigma) \omega([e_{\sigma(0)},e_{\sigma(1)}],e_{\sigma(2)}, \cdots, e_{\sigma(n)} ) \,, </annotation></semantics></math></div> <p>for all <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow> <mi>n</mi></msubsup><mi>Γ</mi><mo stretchy='false'>(</mo><mi>E</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\omega \in \wedge^n_{C^\infty(X)} \Gamma(E)^*</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo>∈</mo><mi>Γ</mi><mo stretchy='false'>(</mo><mi>E</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(e_i \in \Gamma(E))</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Shuff</mi><mo stretchy='false'>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Shuff(p,q)</annotation></semantics></math> denotes the set of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(p,q)</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/shuffle'>shuffle</a>s <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sgn</mi><mo stretchy='false'>(</mo><mi>σ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>sgn(\sigma)</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/signature'>signature</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∈</mo><mo stretchy='false'>{</mo><mo>±</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\in \{\pm 1\}</annotation></semantics></math> of the corresponding <a class='existingWikiWord' href='/nlab/show/diff/permutation'>permutation</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>X = *</annotation></semantics></math> the point we have that <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔞</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{a}</annotation></semantics></math> is a Lie algebra and this definition reproduces the above definition of the CE-algebra of a Lie algebra (possibly up to an irrelevant global sign).</p> <h2 id='of_lie_algebroids_2'>Of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Lie algebroids</h2> <p>See <a class='existingWikiWord' href='/nlab/show/diff/Lie+infinity-algebroid'>∞-Lie algebroid</a>.</p> <h2 id='examples'>Examples</h2> <h3 id='of_abelian_lie_algebras'>Of abelian Lie <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-algebras</h3> <p>The CE-algebra of the Lie algebra of the <a class='existingWikiWord' href='/nlab/show/diff/circle+group'>circle group</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔲</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathfrak{u}(1)</annotation></semantics></math> is the graded-commutative dg-algebra on a single generator in degree 1 with vanishing differential.</p> <p>More generally, the <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebra <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>b</mi> <mi>n</mi></msup><mi>𝔲</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>b^n \mathfrak{u}(1)</annotation></semantics></math> is the one whose CE algebra is the commutative dg-algebra with a single generator in degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math> and vanishing differential.</p> <h3 id='of_'>Of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathfrak{su}(2)</annotation></semantics></math></h3> <p>The CE-algebra of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔰𝔲</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathfrak{su}(2)</annotation></semantics></math> has three generators <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>x, y, z</annotation></semantics></math> in degree one and differential</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><msub><mi>x</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>x</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'> d x_1 = x_2 \wedge x_3 </annotation></semantics></math></div> <p>and cyclically.</p> <h3 id='of_the_tangent_lie_algebroid_'>Of the tangent Lie algebroid <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math></h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a> and <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math> its <a class='existingWikiWord' href='/nlab/show/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a>, the corresponding CE-algebra is the <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham algebra</a> of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>T</mi><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow> <mo>•</mo></msubsup><mi>Γ</mi><mo stretchy='false'>(</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> CE(T X) = (\wedge^\bullet_{C^\infty(X)} \Gamma(T^* X), d_{dR}) \,. </annotation></semantics></math></div> <p>For <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><mi>Γ</mi><mo stretchy='false'>(</mo><mi>T</mi><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(v_i \in \Gamma(T X))</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector field</a>s and <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo>=</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow> <mi>n</mi></msubsup><mi>Γ</mi><mo stretchy='false'>(</mo><mi>T</mi><mi>X</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\omega \in \Omega^n = \wedge^n_{C^\infty(X)} \Gamma(T X)^*</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential form</a> of degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>, the <a href='#OfLieAlgebroids'>formula for the CE-differential</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>d</mi><mi>ω</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>0</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>Sh</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></munder><mi>sgn</mi><mo stretchy='false'>(</mo><mi>σ</mi><mo stretchy='false'>)</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>ω</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>+</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>Shuff</mi><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></munder><mi>sgn</mi><mo stretchy='false'>(</mo><mi>σ</mi><mo stretchy='false'>)</mo><mi>ω</mi><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>]</mo><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> (d\omega)(v_0, \cdots, v_n) = \sum_{\sigma \in Sh(1,n)} sgn(\sigma) v_{\sigma(0)}(\omega(v_{\sigma(1)}, \cdots, v_{\sigma(n)})) + \sum_{\sigma \in Shuff(2,n-1)} sgn(\sigma) \omega([v_{\sigma(0)},v_{\sigma(1)}],v_{\sigma(2)}, \cdots, v_{\sigma(n)} ) \,, </annotation></semantics></math></div> <p>is indeed that for the de Rham differential.</p> <h3 id='of_the_string_lie_2algebra'>Of the string Lie 2-algebra</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> a semisimple Lie algebra with binary <a class='existingWikiWord' href='/nlab/show/diff/invariant+polynomial'>invariant polynomial</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_108' xmlns='http://www.w3.org/1998/Math/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'>\langle -,-\rangle</annotation></semantics></math> – the <a class='existingWikiWord' href='/nlab/show/diff/Killing+form'>Killing form</a> – , the CE-algebra of the <a class='existingWikiWord' href='/nlab/show/diff/String+Lie+2-algebra'>string Lie 2-algebra</a> is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔰𝔱𝔯𝔦𝔫𝔤</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy='false'>(</mo><msup><mi>𝔤</mi> <mo>+</mo></msup><mo>⊕</mo><msup><mi>ℝ</mi> <mo>*</mo></msup><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>d</mi> <mi>string</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> CE(\mathfrak{string}) = (\wedge^\bullet( \mathfrak{g}^+ \oplus \mathbb{R}^*[1]), d_{string}) </annotation></semantics></math></div> <p>where the differential restricted to <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathfrak{g}^*</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>]</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>[-,-]^*</annotation></semantics></math> while on the new generator <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> spanning <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mo>*</mo></msup><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}^*[1]</annotation></semantics></math> it is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mi>b</mi><mo>=</mo><mo stretchy='false'>⟨</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>]</mo><mo stretchy='false'>⟩</mo><mo>∈</mo><msup><mo>∧</mo> <mn>3</mn></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> d b = \langle -, [-,-]\rangle \in \wedge^3 \mathfrak{g}^* \,. </annotation></semantics></math></div> <h3 id='weil_algebra'>Weil algebra</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a>, the CE-algebra of the <a class='existingWikiWord' href='/nlab/show/diff/Lie+2-algebra'>Lie 2-algebra</a> given by the <a class='existingWikiWord' href='/nlab/show/diff/differential+crossed+module'>differential crossed module</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝔤</mi><mover><mo>→</mo><mi>Id</mi></mover><mi>𝔤</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Weil+algebra'>Weil algebra</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>W(\mathfrak{g})</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mover><mo>→</mo><mi>Id</mi></mover><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>W</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> CE(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g}) = W(\mathfrak{g}) \,. </annotation></semantics></math></div> <h3 id='lie_algebra_cohomology'>Lie algebra cohomology</h3> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a> of a <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> with coefficients in the left <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math>-module <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> is defined as <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>H</mi> <mi>Lie</mi> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><mi>𝔤</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo><mo>=</mo><msubsup><mi>Ext</mi> <mrow><mi>U</mi><mi>𝔤</mi></mrow> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><mi>k</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^*_{Lie}(\mathfrak{g},M) = Ext_{U\mathfrak{g}}^*(k,M)</annotation></semantics></math>. It can be computed as <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>𝔤</mi></msub><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Hom_{\mathfrak{g}}(V(\mathfrak{g}),M)</annotation></semantics></math> (a similar story is for <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+homology'>Lie algebra homology</a>) where <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>U</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>⊗</mo><msup><mi>Λ</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>V(\mathfrak{g})=U(\mathfrak{g})\otimes\Lambda^*(\mathfrak{g})</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Chevalley-Eilenberg+chain+complex'>Chevalley-Eilenberg chain complex</a>. If <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> is finite-dimensional over a field then <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>𝔤</mi></msub><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>Λ</mi> <mo>*</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>Hom_{\mathfrak{g}}(V(\mathfrak{g}),k) = CE(\mathfrak{g}) = \Lambda^* \mathfrak{g}^*</annotation></semantics></math> is the underlying complex of the Chevalley-Eilenberg algebra, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/Chevalley-Eilenberg+cochain+complex'>Chevalley-Eilenberg cochain complex</a> with trivial coefficients.</p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a> in degree n of the <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a> of a <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding='application/x-tex'>\mathfrak{g}</annotation></semantics></math> with values in the trivial module <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> is a morphism of <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a>s</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi><mo>→</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>𝔲</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathfrak{g} \to b^{n-1} \mathfrak{u}(1) \,. </annotation></semantics></math></div> <p>In terms of CE-algebras this is a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo><mo>←</mo><mi>CE</mi><mo stretchy='false'>(</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>𝔲</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> CE(\mathfrak{g}) \leftarrow CE(b^{n-1}\mathfrak{u}(1)) \,. </annotation></semantics></math></div> <p>Since by the above example the dg-algebra on he right has a single generator in degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> and vanishing differential, such a morphism is precisely the same thing as a degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-element in <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CE(\mathfrak{g})</annotation></semantics></math>, i.e. an element <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mo>∧</mo> <mi>n</mi></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\omega \in \wedge^n \mathfrak{g}^*</annotation></semantics></math> which is closed under the CE-differential</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><mi>ω</mi><mo>=</mo><mn>0</mn><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> d_{CE} \omega = 0 \,. </annotation></semantics></math></div> <p>This is what one often sees as the definition of a cocycle in <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a>. However, from the general point of view of <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a>, it is better to think of the cocycle equivalently as the morphism <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔤</mi><mo>→</mo><msup><mi>b</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>𝔲</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathfrak{g} \to b^{n-1}\mathfrak{u}(1)</annotation></semantics></math>.</p> <h3 id='brst_complex'>BRST complex</h3> <p>In <a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a>, the Chevalley-Eilenberg algebra <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CE</mi><mo stretchy='false'>(</mo><mi>𝔤</mi><mo>,</mo><mi>N</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CE(\mathfrak{g}, N)</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> of a <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a> or <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a> of a <a class='existingWikiWord' href='/nlab/show/diff/gauge+group'>gauge group</a> <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on space <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> of fields is called the <a class='existingWikiWord' href='/nlab/show/diff/BRST+complex'>BRST complex</a>.</p> <p>In this context</p> <ul> <li> <p>the generators in <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> in degree 0 are called <strong>fields</strong>;</p> </li> <li> <p>the generators <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∈</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\in \mathfrak{g}^*</annotation></semantics></math> in degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> are called <strong>ghosts</strong>;</p> </li> <li> <p>the generators in degree <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math> are called <strong>ghosts of ghosts</strong>;</p> </li> <li> <p>etc.</p> </li> </ul> <p>If <math class='maruku-mathml' display='inline' id='mathml_f786680aa28aabcf0a3f3fa750877173b25ca66d_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> is itself a <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a>, then this is called a <a class='existingWikiWord' href='/nlab/show/diff/BV-BRST+formalism'>BV-BRST complex</a></p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+algebra+cohomology'>∞-Lie algebra cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+L-infinity+algebra'>super L-∞ algebra</a></p> </li> <li> <p><strong>Chevalley-Eilenberg algebra</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative superalgebra</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Weil+algebra'>Weil algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/invariant+polynomial'>invariant polynomial</a></p> </li> </ul> <h2 id='references'>References</h2> <p>An elementary introduction for CE-algebras of Lie algebras is at the beginning of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jos%C3%A9+de+Azc%C3%A1rraga'>José de Azcárraga</a>, J. M. Izquierdo, J. C. Perez Bueno, <em>An introduction to some novel applications of Lie algebra cohomology and physics</em> (<a href='http://arxiv.org/abs/physics/9803046'>arXiv:physics/9803046</a>)</li> </ul> <p>More details are in of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jos%C3%A9+de+Azc%C3%A1rraga'>José de Azcárraga</a>, José M. Izquierdo, section 6.7 of <em><a class='existingWikiWord' href='/nlab/show/diff/Lie+Groups%2C+Lie+Algebras%2C+Cohomology+and+Some+Applications+in+Physics'>Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics</a></em> , Cambridge monographs of mathematical physics, (1995)</li> </ul> <p>See also almost any text on <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+cohomology'>Lie algebra cohomology</a> (see the list of references there).</p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on February 21, 2023 at 17:17:52. 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