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class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+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='#Definition'>Definition</a></li> <li><a href='#history'>History</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>super <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</em> is an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> in the context of <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>: the <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theoretical</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theoretical</a> version of a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>. For more background see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em>.</p> <p>In the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature the <a class="existingWikiWord" href="/nlab/show/formal+duality">formal dual</a> of super <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 of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> came to be known as “<a class="existingWikiWord" href="/nlab/show/FDA">FDA</a>”s (see remark <a class="maruku-ref" href="#SuperLInfintiyAsFDA"></a> below), a decade before plain <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> were discussed in the mathematical literature. The key example in this context are <a class="existingWikiWord" href="/nlab/show/extended+super+Minkowski+spacetimes">extended super Minkowski spacetimes</a>, which are <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a> obtained by iterated <a class="existingWikiWord" href="/nlab/show/universal+higher+central+extension">universal higher central extension</a> from the <a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>. The super-<a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra+cohomology">L-∞ algebra cohomology</a> of these (called “<a class="existingWikiWord" href="/nlab/show/tau-cohomology">tau-cohomology</a>” in the physics literature) turns out to classify <a class="existingWikiWord" href="/nlab/show/super+p-branes">super p-branes</a> and serves as a tool for the construction of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> theories in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a>. For more background on this see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">geometry of physics – fundamental super p-branes</a></em>.</p> <h2 id="Definition">Definition</h2> <p>Abstractly, the definition is immediate:</p> <div class="num_defn" id="SuperLInfinityAlgebra"> <h6 id="definition_2">Definition</h6> <p>A <strong>super <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</strong> is an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (i.e. in <a class="existingWikiWord" href="/nlab/show/chain+complexes+of+super+vector+spaces">chain complexes of super vector spaces</a>).</p> </div> <p>Explicitly this equivalently comes down to the following definition in components:</p> <div class="num_defn" id="SuperGradedSignatureOfPermutation"> <h6 id="definition_3">Definition</h6> <p><strong>(super graded signature of a permutation)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, hence a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times (\mathbb{Z}/2)</annotation></semantics></math>-bigraded vector space.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{v} = (v_1, v_2, \cdots, v_n) </annotation></semantics></math></div> <p>be an <a class="existingWikiWord" href="/nlab/show/n-tuple">n-tuple</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> of homogeneous degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>n</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">(n_i, s_i) \in \mathbb{Z} \times \mathbb{Z}/2</annotation></semantics></math>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>V</mi> <mrow><mo stretchy="false">(</mo><msub><mi>n</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">v_i \in V_{(n_i,s_i)}</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> elements, write <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><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>σ</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{\vert \sigma \vert}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/signature+of+a+permutation">signature of the permutation</a>, which is by definition equal to <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><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^k</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> permutations that each exchange precisely one pair of neighboring elements.</p> <p>We say that the <em>super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{v}</annotation></semantics></math>-graded signature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></em></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><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\} </annotation></semantics></math></div> <p>is the product of the <a class="existingWikiWord" href="/nlab/show/signature+of+a+permutation">signature of the permutation</a> <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><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>σ</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{\vert \sigma \vert}</annotation></semantics></math> with a factor of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mi>i</mi></msub><msub><mi>n</mi> <mi>j</mi></msub></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>s</mi> <mi>i</mi></msub><msub><mi>s</mi> <mi>j</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> (-1)^{n_i n_j}(-1)^{s_i s_j} </annotation></semantics></math></div> <p>for each interchange of neighbours <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>⋯</mi><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>v</mi> <mi>j</mi></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cdots v_i,v_j, \cdots )</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>⋯</mi><msub><mi>v</mi> <mi>j</mi></msub><mo>,</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cdots v_j,v_i, \cdots )</annotation></semantics></math> involved in the decomposition of the permuation as a sequence of swapping neighbour pairs (see at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em> for discussion of this combination of super-grading and homological grading).</p> </div> <p>Now def. <a class="maruku-ref" href="#SuperLInfinityAlgebra"></a> is equivalent to the following def. <a class="maruku-ref" href="#sLInfinityDefinitionViaGeneralizedJacobiIdentity"></a>. This is just the definiton for <a href="#L-infinity-algebra#DefinitionViaHigherBrackets">L-infinity algebras</a>, with the pertinent sign <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> now given by def. <a class="maruku-ref" href="#SuperGradedSignatureOfPermutation"></a>.</p> <div class="num_defn" id="sLInfinityDefinitionViaGeneralizedJacobiIdentity"> <h6 id="definition_4">Definition</h6> <p>An <em>super <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</em> is</p> <ol> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times (\mathbb{Z}/2)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</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>;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/multilinear+map">multilinear map</a>, called the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ary bracket</em>, of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mi>𝔤</mi><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><mi>𝔤</mi></mrow><mo>⏟</mo></munder><mrow><mi>n</mi><mspace width="thickmathspace"></mspace><mtext>copies</mtext></mrow></munder><mo>⟶</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> l_n(\cdots) \;\coloneqq\; [-,-, \cdots, -]_n \;\colon\; \underset{n \; \text{copies}}{\underbrace{\mathfrak{g} \otimes \cdots \otimes \mathfrak{g}}} \longrightarrow \mathfrak{g} </annotation></semantics></math></div> <p>and of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n-2</annotation></semantics></math></p> </li> </ol> <p>such that the following conditions hold:</p> <ol> <li> <p>(<strong>super graded skew symmetry</strong>) each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">l_n</annotation></semantics></math> is graded antisymmetric, in that for every <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> elements and for every <a class="existingWikiWord" href="/nlab/show/n-tuple">n-tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1, \cdots, v_n)</annotation></semantics></math> of homogeneously graded elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>𝔤</mi> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo></mrow></msub></mrow><annotation encoding="application/x-tex">v_i \in \mathfrak{g}_{\vert v_i \vert}</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>n</mi></msub><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><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><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>l</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)}) = \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(\sigma,v_1,\cdots, v_n)</annotation></semantics></math> is the super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1,\cdots,v_n)</annotation></semantics></math>-graded signature of the permuation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#SuperGradedSignatureOfPermutation"></a>;</p> </li> <li> <p>(<strong>strong homotopy <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a></strong>) for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, and for all <a class="existingWikiWord" href="/nlab/show/n-tuple">(n+1)-tuples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1, \cdots, v_{n+1})</annotation></semantics></math> of homogeneously graded elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>𝔤</mi> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo></mrow></msub></mrow><annotation encoding="application/x-tex">v_i \in \mathfrak{g}_{\vert v_i \vert}</annotation></semantics></math> the followig <a class="existingWikiWord" href="/nlab/show/equation">equation</a> holds</p> <div class="maruku-equation" id="eq:LInfinityJacobiIdentity"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>ℕ</mi></mrow></mrow><mrow><mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>UnShuff</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msub><mi>l</mi> <mi>j</mi></msub><mrow><mo>(</mo><msub><mi>l</mi> <mi>i</mi></msub><mrow><mo>(</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>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</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>)</mo></mrow><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}} \sum_{\sigma \in UnShuff(i,j)} \chi(\sigma,v_1, \cdots, v_{n}) (-1)^{i(j-1)} l_{j} \left( l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right), v_{\sigma(i+1)} , \cdots , v_{\sigma(n)} \right) = 0 \,, </annotation></semantics></math></div> <p>where the inner sum runs over all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unshuffles">unshuffles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> is the super graded signature sign from def. <a class="maruku-ref" href="#SuperGradedSignatureOfPermutation"></a>.</p> </li> </ol> <p>A <em>strict <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></em> of super <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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>is</mi><msub><mi>𝔤</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>𝔤</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">is \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2 </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> that preserves the bidegree and all the brackets, in an evident sens.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/homomorphism+of+L-infinity+algebras">strong homotopy homomorphism</a></em> (“<a class="existingWikiWord" href="/nlab/show/sh+map">sh map</a>”) of super <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 is something weaker than that, best defined in <a class="existingWikiWord" href="/nlab/show/formal+duality">formal duals</a>, below in def. <a class="maruku-ref" href="#SuperLInfinityCEAlgebra"></a>.</p> </div> <p>In order to define the correct homomorphisms between super <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 (“sh-maps”) as well as their super-<a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra+cohomology">L-∞ algebra cohomology</a>, consider the following dualization of the above definition:</p> <div class="num_defn" id="SuperLInfinityCEAlgebra"> <h6 id="definition_5">Definition</h6> <p>A super <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> is of <em><a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a></em> if the underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>×</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z} \times (\mathbb{Z}/2)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> is degreewise of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is of finite type, then 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> is the <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> whose underlying <a class="existingWikiWord" href="/nlab/show/graded+algebra">graded algebra</a> is the super-Grassmann algebra</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></mrow><annotation encoding="application/x-tex"> \wedge^\bullet \mathfrak{g}^{\ast} </annotation></semantics></math></div> <p>of the graded degreewise <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^\ast</annotation></semantics></math>, equipped with the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> which on generators is the sum of the <a class="existingWikiWord" href="/nlab/show/dual+linear+maps">dual linear maps</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ary brackets:</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>𝔤</mi></msub><mo>≔</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup><mo lspace="verythinmathspace" rspace="0em">+</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 lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">]</mo> <mo>*</mo></msup><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>⋯</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mo>∧</mo> <mn>1</mn></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>⟶</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> d_{\mathfrak{g}} \coloneqq [-]^\ast + [-,-]^\ast + [-,-,-]^\ast + \cdots \;\colon\; \wedge^1 \mathfrak{g}^\ast \longrightarrow \wedge^\bullet \mathfrak{g}^\ast </annotation></semantics></math></div> <p>and extended 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}^\ast</annotation></semantics></math> as a super-graded <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,even)</annotation></semantics></math>.</p> <p>Notice that here the <a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a> are such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub><mo>∈</mo><msubsup><mi>𝔤</mi> <mrow><mo stretchy="false">(</mo><msub><mi>n</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>s</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">\alpha_i \in \mathfrak{g}^\ast_{(n_i,s_i)}</annotation></semantics></math> elements of homogenous bidegree, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>α</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mn>1</mn></msub><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><msub><mi>s</mi> <mn>1</mn></msub><msub><mi>s</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> \alpha_1 \wedge \alpha_2 \;=\; -(-1)^{n_1 n_2} (-)^{s_1 s_2} </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><msub><mi>d</mi> <mi>𝔤</mi></msub><mo stretchy="false">(</mo><msub><mi>α</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>α</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>α</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∧</mo><msub><mi>α</mi> <mn>2</mn></msub><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><msub><mi>α</mi> <mn>1</mn></msub><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>𝔤</mi></msub><msub><mi>α</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{\mathfrak{g}} (\alpha_1 \wedge \alpha_2) \;=\; (d_{\mathfrak{g}} \alpha_1) \wedge \alpha_2 + (-1)^{n_1} \alpha_1 \wedge (d_{\mathfrak{g}} \alpha_2) \,. </annotation></semantics></math></div> <p>(see at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em> for more on this).</p> <p>A <em>strong homotopy homomorphism</em> (“sh-map”) between super <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>-algbras of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝔤</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>𝔤</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f \;\colon\; \mathfrak{g}_1 \longrightarrow \mathfrak{g}_2 </annotation></semantics></math></div> <p>is defined to be a homomorphism of <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> between their <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> going the other way:</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>𝔤</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟵</mo><mi>CE</mi><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}_1) \longleftarrow CE(\mathfrak{g}_2) \;\colon\; f^\ast </annotation></semantics></math></div> <p>(here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^\ast</annotation></semantics></math> is the primitive concept, and <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> is defined as the <a class="existingWikiWord" href="/nlab/show/formal+duality">formal dual</a> 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>). Hence the <a class="existingWikiWord" href="/nlab/show/category">category</a> of super <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 of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo>↪</mo><msup><mi>dgcsAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> s L_\infty Alg \hookrightarrow dgcsAlg^{op} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebras">differential graded-commutative superalgebras</a> on those that are CE-algebras as above.</p> <p>Finally, the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the <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> of a super <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 of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> is its <em><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra+cohomology">L-∞ algebra cohomology</a></em> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(\mathfrak{g}, \mathbb{R}) \;=\; H^\bullet(CE(\mathfrak{g})) \,. </annotation></semantics></math></div></div> <div class="num_remark" id="LInfinityTerminology"> <h6 id="remark">Remark</h6> <p>Special cases of the general concept of <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a> def. <a class="maruku-ref" href="#sLInfinityDefinitionViaGeneralizedJacobiIdentity"></a> go by special names:</p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is concentrated in even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-degree, it is called an <em><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></em>.</p> <p>If the only possibly non-vanishing brackets 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> are the unary one <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-]</annotation></semantics></math> (which induces the structure of a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> on <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 the binary one, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is equivalently a (super-)<a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is concentrated in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-degrees 0 to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math> then it is called a <em><a class="existingWikiWord" href="/nlab/show/super+Lie+n-algebra">super Lie n-algebra</a></em>.</p> <p>In particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is concentrated in degree 0, then it is equivalently a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>.</p> <p>Combining this, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is concentrated in even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-degree and in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-degree 0 through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math>, then it is called a <em><a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a></em>.</p> <p>In particular if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is concentrated in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-degree 0 and in even <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-degree, then it is equivalently a plain <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>.</p> </div> <h2 id="history">History</h2> <div class="num_remark" id="SuperLInfintiyAsFDA"> <h6 id="remark_2">Remark</h6> <p><strong>(history of the concept of (super-)<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)</strong></p> <p>The identification of the concept of (super-)<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 has a non-linear history:</p> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> in the incarnation of higher brackets satisfying a higher Jacobi identity,</p> <p>def. <a class="maruku-ref" href="#sLInfinityDefinitionViaGeneralizedJacobiIdentity"></a> and remark <a class="maruku-ref" href="#LInfinityTerminology"></a>, were introduced in <a href="https://ncatlab.org/nlab/show/L-infinity-algebra#LadaStasheff92">Lada-Stasheff 92</a>, based on the example of such a structure on the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> that was found in the construction of <a class="existingWikiWord" href="/nlab/show/closed+string+field+theory">closed string field theory</a> in <a href="string+field+theory#Zwiebach93">Zwiebach 92</a>. Some of this history is recalled in <a href="#L-infinity-algebra#Stasheff16">Stasheff 16</a>.</p> <p>The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras (def. <a class="maruku-ref" href="#SuperLInfinityCEAlgebra"></a>) is due to <a href="https://ncatlab.org/nlab/show/L-infinity-algebra#LadaMarkl94">Lada-Markl 94</a>. See <a href="#SatiSchreiberStasheff08">Sati-Schreiber-Stasheff 08, around def. 13</a>.</p> <p>But in this dual incarnation, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> and more generally <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a> (of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a>) had secretly been introduced within the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature already in <a href="#DAuriaFreRegge80">D’Auria-Fré-Regge 80</a> and explicitly in <a href="#Nieuwenhuizen82">van Nieuwenhuizen 82</a>. The concept was picked up in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> (<a href="#DAuriaFre82">D’Auria-Fré 82</a>) and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature (but beware that these dg-algebras are in general <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> only as graded-<a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>, not as differential algebras) The relation between super <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 and the “FDA”s of the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature is made explicit in (<a href="#FSS15">FSS 15</a>).</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Lie+theory">higher Lie theory</a></th><th><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/super+Lie+n-algebra">super Lie n-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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> “FDA” <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> <p>The construction in <a href="#Nieuwenhuizen82">van Nieuwenhuizen 82</a> in turn was motivated by the <a class="existingWikiWord" href="/nlab/show/Sullivan+algebras">Sullivan algebras</a> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> (<a href="#rational+homotopy+theory#Sullivan77">Sullivan 77</a>). Indeed, their dual incarnations in rational homotopy theory are <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> (<a href="#rational+homotopy+theory#Quillen69">Quillen 69</a>), hence a special case 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 (remark <a class="maruku-ref" href="#LInfinityTerminology"></a>)</p> <p>This close relation between <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/higher+Lie+theory">higher Lie theory</a> might have remained less of a secret had it not been for the focus of <a class="existingWikiWord" href="/nlab/show/Sullivan+minimal+models">Sullivan minimal models</a> on the non-<a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> case, which excludes the ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> from the picture. But the Quillen model of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> effectively says that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/rational+topological+space">rational topological space</a> then its <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> is reflected, infinitesimally, by an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>. This perspective began to receive more attention when the <a class="existingWikiWord" href="/nlab/show/Sullivan+construction">Sullivan construction</a> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> was concretely identified as higher <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> in <a href="#Lie+integration#Henriques">Henriques 08</a>. A modern review that makes this <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>-theoretic nature of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> manifest is in <a href="rational+homotopy+theory#BuijsFelixMurillo12">Buijs-Félix-Murillo 12, section 2</a>.</p> </div> <h2 id="properties">Properties</h2> <ul> <li>Every super <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>-Lie algebra has a <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> to a <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a> and a <a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a>. See at <em><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></em> for more on this.</li> </ul> <h2 id="examples">Examples</h2> <p>In the context of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>/<a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a></li> </ul> <p>and its super-<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>-extensions to the</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> (<a class="existingWikiWord" href="/nlab/show/m2brane">m2brane</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/type+II+supergravity+Lie+2-algebra">type II supergravity Lie 2-algebra</a></p> </li> </ul> <p>play a central role. Their exceptional <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">infinity-Lie algebra cohomology</a> governs the consistent <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functionals">Green-Schwarz action functionals</a> for super-<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>-<a class="existingWikiWord" href="/nlab/show/brane">branes</a>. (See the discusson of the <em><a href="Green-Schwarz+action+functional#BraneScan">brane scan</a></em>) there.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">geometry of physics – fundamental super p-branes</a></em> for more on this.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>The <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> might form a super <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 whose brackets give the <a class="existingWikiWord" href="/nlab/show/n-point+function">n-point function</a> of the string, in analogy to what happens for the bosonic string in Zwiebach’s <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a>. (…)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure on chain complexes of super vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> </ul> <h2 id="References">References</h2> <p>In their <a class="existingWikiWord" href="/nlab/show/formal+duality">formal dual</a> incarnations as super-graded commutative <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> (super <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a>), super <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 of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> had secretly been introduced in</p> <ul> <li id="DAuriaFreRegge80"> <p><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a> <a class="existingWikiWord" href="/nlab/show/Tullio+Regge">Tullio Regge</a>, <em>Graded Lie algebra, cohomology and supergravity</em>, Riv. Nuov. Cim. 3, fasc. 12 (1980) (<a href="http://inspirehep.net/record/156191">spire</a>)</p> </li> <li id="Nieuwenhuizen82"> <p><a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <em>Free Graded Differential Superalgebras</em>, in <em>Istanbul 1982, Proceedings, Group Theoretical Methods In Physics</em>, 228-247 and CERN Geneva - TH. 3499 (<a href="http://inspirehep.net/record/182644/">spire</a>)</p> </li> </ul> <p>and hence a whole decade before the explicit appearance of plain (non-super) <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> in <a href="L-infinity-algebra#LadaStasheff92">Lada-Stasheff 92</a>.</p> <p>The concept was picked up in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a></p> <ul> <li id="DAuriaFre82"><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em><a class="existingWikiWord" href="/nlab/files/GeometricSupergravity.pdf" title="Geometric Supergravity in D=11 and its hidden supergroup">Geometric Supergravity in D=11 and its hidden supergroup</a></em>, Nuclear Physics B201 (1982) 101-140 (<a class="existingWikiWord" href="/nlab/files/GeometricSupergravityErrata.pdf" title="errata">errata</a>)</li> </ul> <p>and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature (where in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> one says “<a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a>”, since these dg-algebras are crucially not required to be free as <em>differential</em> algebras).</p> <p>The relation between super <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 and the “FDA”s of the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature is made explicit in:</p> <ul> <li id="FSS15"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, §2 in: <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em>, International Journal of Geometric Methods in Modern Physics <strong>12</strong> 02 (2015) [<a href="https://arxiv.org/abs/1308.5264">arXiv:1308.5264</a>, <a href="http://www.worldscientific.com/doi/abs/10.1142/S0219887815500188">doi:10.1142/S0219887815500188</a>]</p> </li> <li id="FSS18a"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, §2 in: <em><a class="existingWikiWord" href="/schreiber/show/T-Duality+from+super+Lie+n-algebra+cocycles+for+super+p-branes">T-Duality from super Lie n-algebra cocycles for super p-branes</a></em>, Adv. Theor. Math. Phys. <strong>22</strong> 5 (2018) [<a href="https://arxiv.org/abs/1611.06536">arXiv:1611.06536</a>, <a href="https://dx.doi.org/10.4310/ATMP.2018.v22.n5.a3">doi:10.4310/ATMP.2018.v22.n5.a3</a>]</p> </li> <li id="FSS18b"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, (21) in: <em><a class="existingWikiWord" href="/schreiber/show/The+rational+higher+structure+of+M-theory">The rational higher structure of M-theory</a></em>, in: <em>Proceedings of the <a href="http://www.maths.dur.ac.uk/lms/">LMS-EPSRC Durham Symposium</a></em>: <em><a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a></em>, August 2018, Fortschritte der Physik <strong>67</strong> 8-9 (2019) [<a href="https://arxiv.org/abs/1903.02834">arXiv:1903.02834</a>, <a href="https://doi.org/10.1002/prop.201910017">doi:10.1002/prop.201910017</a>]</p> </li> </ul> <p>also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, §3.2 in: <em><a class="existingWikiWord" href="/schreiber/show/Equivariant+homotopy+and+super+M-branes">Real ADE-equivariant (co)homotopy and Super M-branes</a></em>, Comm. Math. Phys. <strong>371</strong> (2019) 425-524 [<a href="https://arxiv.org/abs/1805.05987">arXiv:1805.05987</a>, <a href="https://doi.org/10.1007/s00220-019-03442-3">doi:10.1007/s00220-019-03442-3</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, p. 33, pp. 48 in: <em><a class="existingWikiWord" href="/schreiber/show/Higher+Prequantum+Geometry">Higher Prequantum Geometry</a></em>, in <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, Cambridge University Press (2021) [<a href="https://arxiv.org/abs/1601.05956">arXiv:1601.05956</a>, <a href="https://doi.org/10.1017/9781108854399.008">doi:10.1017/9781108854399.008</a>]</p> </li> </ul> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="geometry+of+physics+--+fundamental+super+p-branes#SuperLInfinityCohomologyAndFDAs">Super L ∞-cohomology and FDAs</a></em>, Section 2 in: <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+fundamental+super+p-branes">geometry of physics – fundamental super p-branes</a></em>, originating in <a class="existingWikiWord" href="/schreiber/show/Super+Lie+n-algebra+of+Super+p-branes">talk notes (2016/17)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Higher+Supergeometry">Introduction to Higher Supergeometry</a></em>, lecture at <em><a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a></em>, <a href="http://www.maths.dur.ac.uk/lms/">Durham Symposium</a> (2019)</p> </li> </ul> <p>Further discussion:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Konstantin+Eder">Konstantin Eder</a>, <a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <a class="existingWikiWord" href="/nlab/show/Simone+Noja">Simone Noja</a>, §16 in: <em>Poincaré Duality for Supermanifolds, Higher Cartan Geometry and Geometric Supergravity</em> [<a href="https://arxiv.org/abs/2312.05224">arXiv:2312.05224</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 27, 2024 at 16:13:03. 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