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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> nonabelian Lie algebra cohomology </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/462/#Item_6" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="lie_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#nonabelian_2cocycles'>Nonabelian 2-cocycles</a></li> <li><a href='#schreiers_theory_for_lie_algebras'>Schreier’s theory for Lie algebras</a></li> <li><a href='#AbstractDefinition'>Abstract definition</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Abstractly, nonabelian Lie algebra cohomology is the restriction of the general notion of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a> to <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>→</mo><mi>der</mi><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g} \to der \mathfrak{h}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">\mathfrak{h}</annotation></semantics></math> are ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>s and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>der</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">der(-)</annotation></semantics></math> denotes the Lie algebra of <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>s.</p> <p>Traditionally abelian <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> is conceived as the cohomology of the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+complex">Chevalley-Eilenberg complex</a> of a Lie algebra and some nonabelian generalizations of this model have been given in the literature. We show below how these definitions are the <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> special cases of the general abstract definition of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a>.</p> <p>The coefficients are not now in a Lie algebra module (which is viewed here as an abelian Lie algebra with action of another Lie algebra), but an arbitrary Lie algebra with something that is action of another Lie algebra up to an inner automorphism.</p> <p>For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of <a class="existingWikiWord" href="/nlab/show/group+extension">factor systems</a>.</p> <p>Below, in the section <a href="#AbstractDefinition">Abstract definition</a> we discuss how a nonabelian Lie algebra cocylce is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\psi,\chi) : \mathfrak{g} \to Der(\mathfrak{k}) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a>s to the <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> of derivations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math>.</p> <p>A generalization (indeed a <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a>) is <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebroid+cohomology">nonabelian Lie algebroid cohomology</a>.</p> <h2 id="nonabelian_2cocycles">Nonabelian 2-cocycles</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> be a field. <strong>Lie algebra factor system</strong> (or a <strong>nonabelian 2-cocycle</strong>) on a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">\mathfrak{b}</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\chi,\psi)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>𝔟</mi><mo>∧</mo><mi>𝔟</mi><mo>→</mo><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\chi: \mathfrak{b}\wedge \mathfrak{b}\to\mathfrak{k}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>𝔟</mi><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi:\mathfrak{b}\to Der(\mathfrak{k})</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-linear maps satisfying</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>χ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>∧</mo><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>∧</mo><mo stretchy="false">[</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>∧</mo><mo stretchy="false">[</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>b</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \chi([b_1,b_2]\wedge b_3)-\chi(b_1\wedge [b_2,b_3])+\chi(b_2\wedge[b_1,b_3]) \\& = \psi(b_3)(\chi(b_1\wedge b_2))-\psi(b_1)(\chi(b_2\wedge b_3))+\psi(b_2)(\chi(b_1\wedge b_3)) \end{aligned} </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>3</mn></msub><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b_1,b_2,b_3\in B</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ad</mi> <mi>𝔨</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>a</mi><mo>∧</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [\psi(a),\psi(b)]=\psi([a,b])+ad_{\mathfrak{k}}(\chi(a\wedge b)) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">a,b\in B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ad</mi> <mi>𝔨</mi></msub><mo>:</mo><mi>𝔨</mi><mo>→</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">ad_{\mathfrak{k}}:\mathfrak{k}\to Int(\mathfrak{k})</annotation></semantics></math> is the canonical map into inner automorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>↦</mo><mo stretchy="false">[</mo><mi>k</mi><mo>,</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k\mapsto [k,]</annotation></semantics></math>.</p> <h2 id="schreiers_theory_for_lie_algebras">Schreier’s theory for Lie algebras</h2> <p><a class="existingWikiWord" href="/nlab/show/Otto+Schreier">Otto Schreier</a> (1926) and Eilenberg-Mac Lane (late 1940-s) developed a theory of nonabelian <a class="existingWikiWord" href="/nlab/show/group+extension">extensions of abstract groups</a> leading to the low dimensional nonabelian group cohomology. For Lie algebras, the theory can be developed in the same manner. One tries to classify <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">extensions of Lie algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>𝔨</mi><mover><mo>→</mo><mi>i</mi></mover><mi>𝔤</mi><mover><mo>→</mo><mi>p</mi></mover><mi>𝔟</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0\to \mathfrak{k} \overset{i}\to \mathfrak{g}\overset{p}\to\mathfrak{b}\to 0 </annotation></semantics></math></div> <p><strong>Theorem.</strong> To every Lie algebra extension as above, and a choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-linear section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>𝔟</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\sigma:\mathfrak{b}\to\mathfrak{g}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, one can assign a nonabelian 2-cocycle (factor system) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">\mathfrak{b}</annotation></semantics></math> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math> as follows: set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>σ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">[</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>σ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\chi(b_1\wedge b_2):=-\sigma([b_1,b_2])+[\sigma(b_1),\sigma(b_2)]</annotation></semantics></math></div> <p>and define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi:\mathfrak{g}\to Der(\mathfrak{k})</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\phi(g)(k):=[g,k]</annotation></semantics></math>. Then set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mo>=</mo><mi>ϕ</mi><mo>∘</mo><mi>σ</mi></mrow><annotation encoding="application/x-tex">\psi:=\phi\circ\sigma</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\chi,\psi)</annotation></semantics></math> is a nonabelian 2-cocycle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">\mathfrak{b}</annotation></semantics></math> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math>.</p> <p><strong>Theorem. (cocycle crossed product of Lie algebras)</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>χ</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\chi,\psi)</annotation></semantics></math> be a factor system as above. Then define a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-linear bracket on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔟</mi><mo>⊕</mo><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{b}\oplus\mathfrak{k}</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mi>χ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>ψ</mi><mo stretchy="false">(</mo><msub><mi>b</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>k</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [(b_1,k_1),(b_2,k_2)] = ([b_1,b_2],\chi(b_1\wedge b_2)+\psi(b_1)(k_2)-\psi(b_2)(k_1)+[k_1,k_2]) </annotation></semantics></math></div> <p>Then</p> <p>(i) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>,</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[,]</annotation></semantics></math> is a antisymmetric and satisfies the Jacobi identity, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>𝔟</mi><mo>⊕</mo><mi>𝔨</mi><mo>,</mo><mo stretchy="false">[</mo><mo>,</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g}:=(\mathfrak{b}\oplus\mathfrak{k},[,])</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-Lie algebra.</p> <p>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>↦</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k\mapsto (0,k)</annotation></semantics></math> defines an embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>𝔨</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">i:\mathfrak{k}\to\mathfrak{g}</annotation></semantics></math> of Lie algebras and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">(b,k)\mapsto b</annotation></semantics></math> is a surjective homomorphism of Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">p:\mathfrak{g}\to\mathfrak{b}</annotation></semantics></math> whose kernel is the Lie ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>⊕</mo><mi>𝔨</mi><mo>⊂</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">i(\mathfrak{k})=0\oplus\mathfrak{k}\subset\mathfrak{g}</annotation></semantics></math>. This way <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>𝔨</mi><mover><mo>→</mo><mi>i</mi></mover><mi>𝔤</mi><mover><mo>→</mo><mi>p</mi></mover><mi>𝔟</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\to\mathfrak{k}\overset{i}\to\mathfrak{g}\overset{p}\to\mathfrak{b}\to 0</annotation></semantics></math> is an extension of the base Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔟</mi></mrow><annotation encoding="application/x-tex">\mathfrak{b}</annotation></semantics></math> by the kernel Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math>.</p> <p>(iii) If the 2-cocycle is obtained from a Lie algebra extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>𝔨</mi><mover><mo>→</mo><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mover><msub><mi>𝔤</mi> <mn>0</mn></msub><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mover><mi>𝔟</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">0\to \mathfrak{k}\overset{i_0}\to \mathfrak{g}_0\overset{p_0}\to\mathfrak{b}\to 0</annotation></semantics></math> and an arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-linear section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_0</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">p_0</annotation></semantics></math>, then the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>can</mi> <mi>σ</mi></msub><mo>:</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">can_\sigma:\mathfrak{g}_0\to\mathfrak{g}</annotation></semantics></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g\mapsto (p(g),-\sigma(p(g))+g)</annotation></semantics></math> is well-defined and a Lie algebra isomorphism such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>can</mi> <mi>σ</mi></msub><mo>∘</mo><msub><mi>i</mi> <mn>0</mn></msub><mo>=</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">can_\sigma\circ i_0=i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub><mo>=</mo><mi>p</mi><mo>∘</mo><msub><mi>can</mi> <mi>σ</mi></msub></mrow><annotation encoding="application/x-tex">p_0=p\circ can_\sigma</annotation></semantics></math>, hence the two extensions are isomorphic.</p> <p>In addition to the problem of extensions, nonabelian 2-cocycles appear in a more general problem of liftings of Lie algebras.</p> <h2 id="AbstractDefinition">Abstract definition</h2> <p>We claim that the above definition of nonabelian Lie algebra cocycles may be understood naturally in terms of the general notion of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> and in particular is the image of the story of <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a> under Lie differentiation:</p> <div class="standout"> <p>The following observation is not in the literature.</p> </div> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Lie</mi></mrow><annotation encoding="application/x-tex">\infty Lie</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a>s. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>,</mo><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}, \mathfrak{k}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>s. Then the degree 2 nonabelian Lie algebra cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>nonab</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mn>∞</mn><mi>Lie</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> H^2_{nonab}(\mathfrak{g}, \mathfrak{k}) \simeq \pi_0 \infty Lie(\mathfrak{g}, Der(\mathfrak{k})) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Der(\mathfrak{k})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> of derivations on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math>.</p> <p>More in detail:</p> <ul> <li> <p>nonabelian degree 2 Lie algebra cocycles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi,\xi)</annotation></semantics></math> are in natural bijections with morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{g} \to Der(\mathfrak{k}) </annotation></semantics></math></div></li> <li> <p>coboundaries <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> between cocycles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ψ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ξ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi_1,\xi_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ψ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ξ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\psi_2,\xi_2)</annotation></semantics></math> correspond to homotopies between these</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ψ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>χ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd> <mtd><msup><mo>⇓</mo> <mi>η</mi></msup></mtd> <mtd><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msup><mo>↗</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ψ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>χ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow\searrow^{\mathrlap{(\psi_1,\chi_1)}} \\ \mathfrak{g} &\Downarrow^{\eta}& Der(\mathfrak{k}) \\ & \searrow\nearrow^{\mathrlap{(\psi_2,\chi_2)}} } </annotation></semantics></math></div> <p>and this correspondence is precise if we take the homotopy to be induced from the “standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>”, described below.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Checking this is a straightforward matter of unwinding the definitions of morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>Which is what we indicate.</p> <p>We model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Lie</mi></mrow><annotation encoding="application/x-tex">\infty Lie</annotation></semantics></math> as usual a subcategory of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s of <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a>s, by representing each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> by its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>.</p> <p>For the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> itself with Lie bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>𝔤</mi><mo>∧</mo><mi>𝔤</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}</annotation></semantics></math> this is the <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</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><mspace width="thickmathspace"></mspace><mi>d</mi><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 stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) = (\wedge^\bullet \mathfrak{g}^* , \; d = [-,-]^* ) \,, </annotation></semantics></math></div> <p>where the differential is on generators the dual of the Lie bracket, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><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">[-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^* </annotation></semantics></math> extended as a graded derivation to all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> <p>For any <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> coming from a <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝔥</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>𝔥</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{h}_1 \stackrel{\delta}{\to} \mathfrak{h}_1)</annotation></semantics></math> with action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mi>𝔥</mi> <mn>1</mn></msub><mo>→</mo><mi>der</mi><mo stretchy="false">(</mo><msub><mi>𝔥</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho : \mathfrak{h}_1 \to der(\mathfrak{h}_2)</annotation></semantics></math> – that we think of in the following as equivalently a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mi>𝔥</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>𝔥</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>𝔥</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\rho : \mathfrak{h}_1 \otimes \mathfrak{h}_2 \to \mathfrak{h}_2</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><msub><mi>𝔥</mi> <mn>2</mn></msub><mover><mo>→</mo><mi>δ</mi></mover><msub><mi>𝔥</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊕</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>δ</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{h}_2 \stackrel{\delta}{\to} \mathfrak{h}_1) = \left( \wedge^\bullet ( \mathfrak{h}_1^* \oplus \mathfrak{h}_2^* ) , \; d_{\delta} \right) </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">\mathfrak{h}_1^*</annotation></semantics></math> in degree 1 and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">\mathfrak{h}_2^*</annotation></semantics></math> in degree 2, and with the differential given on degree 1 generators by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>δ</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msubsup><mo stretchy="false">]</mo> <mrow><msub><mi>𝔥</mi> <mn>1</mn></msub></mrow> <mo>*</mo></msubsup><mo lspace="verythinmathspace" rspace="0em">+</mo><msup><mi>δ</mi> <mo>*</mo></msup><mo>:</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>→</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>∧</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊕</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex"> d_\delta |_{\mathfrak{h}_1^*} = [-,-]_{\mathfrak{h}_1}^* + \delta^* : \mathfrak{h}_1^* \to \mathfrak{h}_1^* \wedge \mathfrak{h}_1^* \oplus \mathfrak{h}_2^* </annotation></semantics></math></div> <p>and on degree 2 generators by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>δ</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow></msub><mo>=</mo><msup><mi>ρ</mi> <mo>*</mo></msup><mo>:</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup><mo>→</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊗</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_\delta |_{\mathfrak{h}_2^*} = \rho^* : \mathfrak{h}_2^* \to \mathfrak{h}_1^* \otimes \mathfrak{h}_2^* \,. </annotation></semantics></math></div> <p>The case of the derivation <a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a> of a Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔨</mi></mrow><annotation encoding="application/x-tex">\mathfrak{k}</annotation></semantics></math> is the special case of this for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>𝔨</mi><mover><mo>→</mo><mi>ad</mi></mover><mi>der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Der(\mathfrak{k}) = (\mathfrak{k} \stackrel{ad}{\to} der(\mathfrak{k})) \,. </annotation></semantics></math></div> <p>Now a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\psi, \chi) : \mathfrak{g} \to Der(\mathfrak{k}) </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras is given by a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><msup><mi>ψ</mi> <mo>*</mo></msup><mo>,</mo><msup><mi>χ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) \leftarrow CE(Der(\mathfrak{k})) : (\psi^*, \chi^*) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s.</p> <p>Morphisms of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s are given by morphisms of the underlying graded algebras, subject to the respect for the differentials. Morphisms of the underlying graded <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a>s are given by grading preserving linear maps on the spaces of generators.</p> <p>So the underlying maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>←</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊕</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><msup><mi>ψ</mi> <mo>*</mo></msup><mo>,</mo><msup><mi>χ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \wedge^\bullet \mathfrak{g}^* \leftarrow \wedge^\bullet (\mathfrak{h}_1^* \oplus \mathfrak{h}_2^*) : (\psi^* , \chi^*) </annotation></semantics></math></div> <p>come from linear maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>←</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>:</mo><msup><mi>ψ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \mathfrak{g}^* \leftarrow \mathfrak{h}_1^* : \psi^* </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>←</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup><mo>:</mo><msup><mi>χ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> \mathfrak{g}^* \wedge \mathfrak{g}^* \leftarrow \mathfrak{h}_2^* : \chi^* </annotation></semantics></math></div> <p>i.e. form linear maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \psi : \mathfrak{g} \to der(\mathfrak{k}) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>𝔤</mi><mo>∧</mo><mi>𝔤</mi><mo>→</mo><mi>𝔨</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \chi : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{k} \,. </annotation></semantics></math></div> <p>This is the underlying data of the nonabelian 2-cocycle. Now the respect for the differentials on the Chevalley-Eilenberg algebras will give the cocycle condition:</p> <p>let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>𝔥</mi> <mn>2</mn> <mo>*</mo></msubsup><mo>⊂</mo><mi>CE</mi><mo stretchy="false">(</mo><msub><mi>𝔥</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝔥</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \mathfrak{h}_2^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2)</annotation></semantics></math> be any degree 2 element, then respect for the differential implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ω</mi><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>χ</mi><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 stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><msup><mi>ψ</mi> <mo>*</mo></msup><mo>,</mo><msup><mi>χ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>ω</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msub><mi>d</mi> <mi>𝔤</mi></msub><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></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msub><mi>d</mi> <mi>δ</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ω</mi><mo stretchy="false">(</mo><mi>ξ</mi><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></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><msup><mi>ψ</mi> <mo>*</mo></msup><mo>,</mo><msup><mi>χ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>ω</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \omega(\chi([-,-],-)) = \omega(\rho(\psi(-)(\chi(-,-)))) &\stackrel{(\psi^*, \chi^*)}{\leftarrow}& \omega(\rho(-)(-))) \\ \uparrow^{d_\mathfrak{g} = [-,-]^*} && \uparrow^{\mathrlap{d_{\delta}}} \\ \omega(\xi(-,-)) &\stackrel{(\psi^*, \chi^*)}{\leftarrow}& \omega } \,. </annotation></semantics></math></div> <p>Since this has to hold for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, we get the first part of the cocycle condition:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mi>χ</mi><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></mrow><annotation encoding="application/x-tex"> \chi([-,-],-) = \rho(\psi(-)\chi(-,-)) </annotation></semantics></math></div> <p>(both sides here regarded as elements of a graded Grassmann algebra as indicated above, so with all antisymmetrization on the arguments implicit).</p> <p>Similarly, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>∈</mo><msubsup><mi>𝔥</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊂</mo><mi>CE</mi><mo stretchy="false">(</mo><msub><mi>𝔥</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝔥</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda \in \mathfrak{h}_1^* \subset CE(\mathfrak{h}_1 \to \mathfrak{h}_2)</annotation></semantics></math> be any degree 1 element, then respect for the differential implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>λ</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</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 stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>+</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>ad</mi><mo stretchy="false">(</mo><mi>χ</mi><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 stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow></mrow></mover></mtd> <mtd><mi>λ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mrow><msub><mi>𝔥</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>ad</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>λ</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><msup><mi>ψ</mi> <mo>*</mo></msup><mo>,</mo><msup><mi>χ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>λ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \lambda(\psi([-,-])) = \lambda([\psi(-), \psi(-)]) + \lambda(ad(\chi(-,-))) &\stackrel{}{\leftarrow}& \lambda([-,-]_{\mathfrak{h}_1}) + \lambda(ad(-)) \\ \uparrow && \uparrow \\ \lambda(\psi(-)) &\stackrel{(\psi^* , \chi^*)}{\leftarrow}& \lambda } \,. </annotation></semantics></math></div> <p>Again, this has to hold for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>, so we have the auxiliary condition on the cocycle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo stretchy="false">(</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><mo stretchy="false">[</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>ψ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>+</mo><mi>ad</mi><mo stretchy="false">(</mo><mi>ξ</mi><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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \psi([-,-]) = [\psi(-),\psi(-)] + ad(\xi(-,-)) \,. </annotation></semantics></math></div> <p>This shows that morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>→</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{g} \to Der(\mathfrak{k})</annotation></semantics></math> are in bijection to the nonabelian cocycles.</p> <p>It remains to show that the homotopies map to coboundaries. For that we may take in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Lie</mi></mrow><annotation encoding="application/x-tex">\infty Lie</annotation></semantics></math> the standard <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a> of some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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>𝔤</mi><mo stretchy="false">)</mo><mo>←</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>←</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g})\otimes CE(\mathfrak{g}) \leftarrow C^\bullet(\Delta^1)\otimes CE(\mathfrak{g}) \leftarrow CE(\mathfrak{g}) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\bullet(\Delta^1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a> of cochains on the cellular 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>, i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">⟩</mo><mo>⊕</mo><mo stretchy="false">⟨</mo><mi>c</mi><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mi>a</mi><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>d</mi><mi>b</mi><mo>=</mo><mi>d</mi><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C^\bullet(\Delta^1) = (\wedge^\bullet (\langle a,b \rangle \oplus \langle c \rangle) , d a = - d b = d c ) \,, </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a,b</annotation></semantics></math> generators in degree 0 and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> in degree 1. Using this, write out the data implied by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> that is a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↖</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>ψ</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>,</mo><msubsup><mi>χ</mi> <mn>1</mn> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>←</mo><mi>η</mi></mover></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>Der</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msubsup><mi>ψ</mi> <mn>2</mn> <mo>*</mo></msubsup><mo>,</mo><msubsup><mi>χ</mi> <mn>2</mn> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CE(\mathfrak{g}) \\ \downarrow & \nwarrow^{\mathrlap{(\psi_1^*, \chi_1^*)}} \\ C^\bullet(\Delta^1)\otimes CE(\mathfrak{g}) &\stackrel{\eta}{\leftarrow}& CE(Der(\mathfrak{k})) \\ \uparrow & \swarrow_{\mathrlap{(\psi_2^*, \chi_2^*)}} \\ CE(\mathfrak{g}) } </annotation></semantics></math></div> <p>along the above lines.</p> <p>Notice that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>dgAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">dgAlg^{op}</annotation></semantics></math> every object is cofibrant, so that this is indeed a left homotopy. See <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a> for more on this.</p> </div> <h2 id="references">References</h2> <p>On original source is</p> <ul> <li>G. Hochschild, <em>Lie algebra kernels and cohomology</em>, Amer. J. Math. <strong>76</strong>, n.3 (1954) 698–716.</li> </ul> <p>The notation/exposition in sections 2 and 3 is adapted from personal notes of Z. Škoda (1997). A systematic theory has been many times partly rediscovered from soon after the Eilenberg–Mac Lane work on group extension, among first by Hochschild and then by many others till nowdays. Here is a recent online account emphasising parallels with <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitri+Alekseevsky">Dmitri Alekseevsky</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Michor">Peter Michor</a>, Wolfgang Ruppert, Extensions of Lie algebras (<a href="http://arxiv.org/abs/math/0005042">math.DG/0005042</a>)</li> </ul> <p>A more conceptual picture is in a work of <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a> which extends also to its <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>, extensions of <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a>s. See</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, Lie 2-algebras and the geometry of gerbes, Unni Namboodiri Lectures 2006 <a href="http://math.ucr.edu/home/baez/namboodiri/stevenson_maclane.pdf">slides</a></li> </ul> <p>There is also</p> <ul> <li>N. Inassaridze, E. Khmaladze, M. Ladra, <em>Non-abelian cohomology and extensions of Lie algebras</em> Journal of Lie Theory <strong>18</strong> (2008) 413–432 (<a href="http://www.rmi.acnet.ge/~khmal/PAPERS/LieTheory2008.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 2, 2024 at 10:25:43. 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