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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> <h4 id="superalgebra_and_supergeometry">Super-Algebra and Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <ul> <li> <p><a 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> </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> <ul> <li><a href='#AsLieAlgebrasInternalToSuperVectorSpaces'>As Lie algebras internal to super vector spaces</a></li> <li><a href='#as_supergraded_lie_algebras'>As super-graded Lie algebras</a></li> <li><a href='#as_formal_duals_of_a_chevalleyeilenberg_superalgebras'>As formal duals of a Chevalley-Eilenberg super-algebras</a></li> <li><a href='#AsSuperRepresentableLieAlgebrasOverSuperpoints'>As super-representable Lie algebras in the topos over superpoints</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#Classification'>Classification</a></li> <li><a href='#RelationToDGLieAlgebras'>Relation to dg-Lie algebras</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#basic_examples'>Basic examples</a></li> <li><a href='#superpoincar_super_lie_algebras_supersymmetry'>Super-Poincaré super Lie algebras (supersymmetry)</a></li> <li><a href='#SuperLieAlgebraInducedFromVectorSpace'>Embedding tensors and tensor hierarchy</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>super Lie algebra</em> is the analog of a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> in <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>/<a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>.</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></em>.</p> <h2 id="definition">Definition</h2> <p>There are various equivalent ways to state the definition of super Lie algebras. Here are a few (for more discussion see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em>):</p> <h3 id="AsLieAlgebrasInternalToSuperVectorSpaces">As Lie algebras internal to super vector spaces</h3> <div class="num_defn" id="SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces"> <h6 id="definition_2">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></em> is a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie algebra object</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi><mo>=</mo><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo>,</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><msup><mi>τ</mi> <mi>super</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sVect = (Vect^{\mathbb{Z}/2}, \otimes_k, \tau^{super} )</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (a <em><a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie algebra object</a> in <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a></em>). Hence this is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super 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>a homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi><mo>⟶</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> [-,-] \;\colon\; \mathfrak{g} \otimes_k \mathfrak{g} \longrightarrow \mathfrak{g} </annotation></semantics></math></div> <p>of super vector spaces (the <em>super Lie bracket</em>)</p> </li> </ol> <p>such that</p> <ol> <li> <p>the bracket is skew-symmetric in that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>τ</mi> <mrow><mi>𝔤</mi><mo>,</mo><mi>𝔤</mi></mrow> <mi>super</mi></msubsup></mrow></mover></mtd> <mtd><mi>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></munder></mtd> <mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathfrak{g} \otimes_k \mathfrak{g} & \overset{\tau^{super}_{\mathfrak{g},\mathfrak{g}}}{\longrightarrow} & \mathfrak{g} \otimes_k \mathfrak{g} \\ {}^{\mathllap{[-,-]}}\downarrow && \downarrow^{\mathrlap{[-,-]}} \\ \mathfrak{g} &\underset{-1}{\longrightarrow}& \mathfrak{g} } </annotation></semantics></math></div> <p>(here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>τ</mi> <mi>super</mi></msup></mrow><annotation encoding="application/x-tex">\tau^{super}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> in the category of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a>)</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> holds in that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>τ</mi> <mrow><mi>𝔤</mi><mo>,</mo><mi>𝔤</mi></mrow> <mi>super</mi></msubsup><msub><mo>⊗</mo> <mi>k</mi></msub><mi>id</mi></mrow></mover></mtd> <mtd></mtd> <mtd><mi>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mpadded width="0" lspace="-100%width"><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow><mo>]</mo></mrow></mpadded><mo>−</mo><mrow><mo>[</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow><mo>]</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g} && \overset{\tau^{super}_{\mathfrak{g}, \mathfrak{g}} \otimes_k id }{\longrightarrow} && \mathfrak{g} \otimes_k \mathfrak{g} \otimes_k \mathfrak{g} \\ & {}_{\mathllap{\left[-,\left[-,-\right]\right]} - \left[\left[-,-\right],-\right] }\searrow && \swarrow_{\mathrlap{\left[-,\left[-,-\right]\right]}} \\ && \mathfrak{g} } \,. </annotation></semantics></math></div></li> </ol> </div> <h3 id="as_supergraded_lie_algebras">As super-graded Lie algebras</h3> <p>Externally this means the following:</p> <div class="num_prop" id="SuperLieAlgebraTraditional"> <h6 id="proposition">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> according to def. <a class="maruku-ref" href="#SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces"></a> is equivalently</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 stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>even</mi></msub><mo>⊕</mo><msub><mi>𝔤</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}</annotation></semantics></math>;</p> </li> <li> <p>equipped with a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> (the <em>super Lie bracket</em>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>𝔤</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝔤</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> [-,-] : \mathfrak{g}\otimes_k \mathfrak{g} \to \mathfrak{g} </annotation></semantics></math></div> <p>which is <em>graded</em> skew-symmetric: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x,y \in \mathfrak{g}</annotation></semantics></math> two elements of homogeneous degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\sigma_y</annotation></semantics></math>, respectively, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mi>x</mi></msub><msub><mi>σ</mi> <mi>y</mi></msub></mrow></msup><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [x,y] = -(-1)^{\sigma_x \sigma_y} [y,x] \,, </annotation></semantics></math></div></li> <li> <p>that satisfies the <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>-graded <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> in that for any three elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x,y,z \in \mathfrak{g}</annotation></semantics></math> of homogeneous super-degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>x</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>y</mi></msub><mo>,</mo><msub><mi>σ</mi> <mi>z</mi></msub><mo>∈</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_x,\sigma_y,\sigma_z\in \mathbb{Z}_2</annotation></semantics></math> then</p> <div class="maruku-equation" id="eq:GradedJacobiIdentity"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mi>x</mi></msub><mo>⋅</mo><msub><mi>σ</mi> <mi>y</mi></msub></mrow></msup><mo stretchy="false">[</mo><mi>y</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [x, [y, z] ] = [ [x,y],z] + (-1)^{\sigma_x \cdot \sigma_y} [y, [x,z] ] \,. </annotation></semantics></math></div></li> </ol> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of super Lie algebras is a homomorphisms of the underlying <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> which preserves the Lie bracket. We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sLieAlg</mi></mrow><annotation encoding="application/x-tex"> sLieAlg </annotation></semantics></math></div> <p>for the resulting <a class="existingWikiWord" href="/nlab/show/category">category</a> of super Lie algebras.</p> </div> <h3 id="as_formal_duals_of_a_chevalleyeilenberg_superalgebras">As formal duals of a Chevalley-Eilenberg super-algebras</h3> <div class="num_defn" id="CEAlgebraofSuperLieAlgebra"> <h6 id="definition_3">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, then its <em><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></em> <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 super-<a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> on the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual</a> super vector space</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>equipped with a <a class="existingWikiWord" href="/nlab/show/differential">differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>𝔤</mi></msub></mrow><annotation encoding="application/x-tex">d_{\mathfrak{g}}</annotation></semantics></math> that on generators is the linear dual of the super Lie bracket</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><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><msup><mo stretchy="false">]</mo> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>∧</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> d_{\mathfrak{g}} \;\coloneqq\; [-,-]^\ast \;\colon\; \mathfrak{g}^\ast \to \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast </annotation></semantics></math></div> <p>and which is extended to <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> by the graded Leibniz rule (i.e. as a graded <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>).</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>Here all elements are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℤ</mi><mo>×</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, the first being the <em>cohomological grading</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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo lspace="0em" rspace="thinmathspace">n</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\n \mathfrak{g}^\ast</annotation></semantics></math>, the second being the <em>super-grading</em> <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> (even/odd).</p> <p>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><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_i \in CE(\mathfrak{g})</annotation></semantics></math> two elements of homogeneous bi-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>σ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n_i, \sigma_i)</annotation></semantics></math>, respectively, the <a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">sign rule</a> is</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><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><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub></mrow></msup><mspace width="thickmathspace"></mspace><msub><mi>α</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>α</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \alpha_1 \wedge \alpha_2 = (-1)^{n_1 n_2} (-1)^{\sigma_1 \sigma_2}\; \alpha_2 \wedge \alpha_1 \,. </annotation></semantics></math></div> <p>(See at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em> for discussion of this sign rule and of an alternative sign rule that is also in use. )</p> </div> <p>We may think of <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> equivalently as the <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> of <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">left-invariant</a> <a class="existingWikiWord" href="/nlab/show/super+differential+forms">super differential forms</a> on the <a class="existingWikiWord" href="/nlab/show/Lie+theory">corresponding</a> simply connected <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> .</p> <p>The concept of <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> is traditionally introduced as a means to define <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>:</p> <div class="num_defn" id="SuperLieAlgebraCohomology"> <h6 id="definition_4">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super 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>, then</p> <ol> <li> <p>an <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>-cocycle</em> 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> (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>) is an element of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,even)</annotation></semantics></math> in 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> (def. <a class="maruku-ref" href="#CEAlgebraofSuperLieAlgebra"></a>) which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>𝕘</mi></msub></mrow><annotation encoding="application/x-tex">d_{\mathbb{g}}</annotation></semantics></math> closed.</p> </li> <li> <p>the cocycle is non-trivial if it is not <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>𝔤</mi></msub></mrow><annotation encoding="application/x-tex">d_{\mathfrak{g}}</annotation></semantics></math>-exact</p> </li> <li> <p>hene the <em>super-<a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></em> 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 <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>) is the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></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><mo>=</mo><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></li> </ol> </div> <p>The following says that the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> is an equivalent incarnation of the <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>sLieAlg</mi> <mi>fin</mi></msup><mo>↪</mo><msup><mi>dgAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> CE \;\colon\; sLieAlg^{fin} \hookrightarrow dgAlg^{op} </annotation></semantics></math></div> <p>that sends a finite dimensional <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super 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> to 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> (def. <a class="maruku-ref" href="#CEAlgebraofSuperLieAlgebra"></a>) is a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a> which hence exibits <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> as a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/differential-graded+algebras">differential-graded algebras</a>.</p> </div> <h3 id="AsSuperRepresentableLieAlgebrasOverSuperpoints">As super-representable Lie algebras in the topos over superpoints</h3> <p>Equivalently, a super Lie algebra is a “super-representable” <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie algebra object</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Super%E2%88%9EGrpd">Super∞Grpd</a> over the site of <a class="existingWikiWord" href="/nlab/show/super+points">super points</a> (<a href="#Sachse08">Sachse 08, Section 3.2, towards cor. 3.3</a>).</p> <p>See the discussion at <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> for details on this.</p> <h2 id="properties">Properties</h2> <h3 id="Classification">Classification</h3> <p>(<a href="Kac77a">Kac 77a</a>, <a href="#Kac77b">Kac 77b</a>) states a classification of super Lie algebras which are</p> <ol> <li> <p>finite dimensional</p> </li> <li> <p>simple</p> </li> <li> <p>over a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>.</p> </li> </ol> <p>Such an algebra is called of <em>classical type</em> if the <a class="existingWikiWord" href="/nlab/show/action">action</a> of its even-degree part on the odd-degree part is completely reducible. Those simple finite dimensional algebras not of classical type are of <em>Cartan type</em>.</p> <ol> <li> <p>classical type</p> <ol> <li> <p>four infinite series</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(m,n)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">B(m,n) = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2m+1,2n)</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">m\geq 0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \gt 0</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(n)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo></mrow><annotation encoding="application/x-tex">D(m,n) = </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/osp">osp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>m</mi><mo>,</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2m,2n)</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">m \geq 2</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \gt 0</annotation></semantics></math></p> </li> </ol> </li> <li> <p>two exceptional ones</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(4)</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G%283%29">G(3)</a></p> </li> </ol> </li> <li> <p>a family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>;</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(2,1;\alpha)</annotation></semantics></math> of deformations of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(2,1)</annotation></semantics></math></p> </li> <li> <p>two “strange” series</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(n)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(n)</annotation></semantics></math></p> </li> </ol> </li> </ol> </li> <li> <p>Cartan type</p> <p>(…)</p> </li> </ol> <p>The underlying even-graded <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> for type 2 is as follows</p> <table><thead><tr><th><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></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{even}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{even}</annotation></semantics></math> rep on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{odd}</annotation></semantics></math></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><mi>B</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B(m,n)</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><msub><mi>B</mi> <mi>m</mi></msub><mo>⊕</mo><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">B_m \oplus C_n</annotation></semantics></math></td><td style="text-align: left;">vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(m,n)</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><msub><mi>D</mi> <mi>m</mi></msub><mo>⊕</mo><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">D_m \oplus C_n</annotation></semantics></math></td><td style="text-align: left;">vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(2,1,\alpha)</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><msub><mi>A</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1 \oplus A_1 \oplus A_1</annotation></semantics></math></td><td style="text-align: left;">vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(4)</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><msub><mi>B</mi> <mn>3</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">B_3\otimes A_1</annotation></semantics></math></td><td style="text-align: left;">spinor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(3)</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><msub><mi>G</mi> <mn>2</mn></msub><mo>⊕</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">G_2\oplus A_1</annotation></semantics></math></td><td style="text-align: left;">spinor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(n)</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><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">A_n</annotation></semantics></math></td><td style="text-align: left;">adjoint</td></tr> </tbody></table> <p>For type 1 the <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><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-grading lifts to an <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>-grading with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>=</mo><msub><mi>𝔤</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo>⊕</mo><msub><mi>𝔤</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1</annotation></semantics></math>.</p> <table><thead><tr><th><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></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{even}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{even}</annotation></semantics></math> rep on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_{{-1}}</annotation></semantics></math></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><mi>A</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(m,n)</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><msub><mi>A</mi> <mi>m</mi></msub><mo>⊕</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>⊕</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">A_m \oplus A_n \oplus \mathbb{C}</annotation></semantics></math></td><td style="text-align: left;">vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> <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">\mathbb{C}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>m</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(m,m)</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><msub><mi>A</mi> <mi>m</mi></msub><mo>⊕</mo><msub><mi>A</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">A_m \oplus A_m</annotation></semantics></math></td><td style="text-align: left;">vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> vector</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(n)</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><msub><mi>C</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">C_{n-1} \oplus \mathbb{C}</annotation></semantics></math></td><td style="text-align: left;">vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> <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">\mathbb{C}</annotation></semantics></math></td></tr> </tbody></table> <p>reviewed e.g. in (<a href="#Farmer84">Farmer 84, p. 25,26</a>, <a href="#Minwalla98">Minwalla 98, section 4.1</a>).</p> <h3 id="RelationToDGLieAlgebras">Relation to dg-Lie algebras</h3> <p>A <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mo>∂</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g}, \partial, [-,-])</annotation></semantics></math> may be understood equivalently as a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔤</mi><mo>=</mo><msub><mi>𝔤</mi> <mi>even</mi></msub><mo>⊕</mo><msub><mi>𝔤</mi> <mi>odd</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd}, [-,-,]) </annotation></semantics></math></div> <p>equipped with</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">grading</a> of the underlying <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> to 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 href="graded+vector+space#SpecialCaseOfZGradedVectorSpaces">-graded vector space</a> through the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo>=</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} = \mathbb{Z}_2</annotation></semantics></math>, hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mi>even</mi></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mo>⊕</mo><mi>n</mi></munder><msub><mi>𝔤</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><mphantom><mi>AAA</mi></mphantom><msub><mi>𝔤</mi> <mi>odd</mi></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><munder><mo>⊕</mo><mi>n</mi></munder><msub><mi>𝔤</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathfrak{g}_{even} \;\simeq\; \underset{n }{\oplus} \mathfrak{g}_{2n} \,, \phantom{AAA} \mathfrak{g}_{odd} \;\simeq\; \underset{n }{\oplus} \mathfrak{g}_{2n+1} </annotation></semantics></math></div> <p>such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝔤</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msub><mo>⊗</mo><msub><mi>𝔤</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo>⟶</mo><msub><mi>𝔤</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> [-,-] \;\colon\; \mathfrak{g}_{n_1} \otimes \mathfrak{g}_{n_2} \longrightarrow \mathfrak{g}_{n_1 + n_2} </annotation></semantics></math></div></li> <li> <p>an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>∈</mo><msub><mi>𝔤</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Q \in \mathfrak{g}_{-1}</annotation></semantics></math></p> <p>such that</p> <div class="maruku-equation" id="eq:NilpotencyOfQ"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> [Q,Q] = 0 </annotation></semantics></math></div></li> </ol> <p>(See also at “<a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a>”.)</p> <p>Given this, define the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> to be the <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial \;\coloneqq\; [Q,-] \,. </annotation></semantics></math></div> <p>That this differential squares to 0 follows by the super-Jacobi identity <a class="maruku-eqref" href="#eq:GradedJacobiIdentity">(1)</a> and by the nilpotency <a class="maruku-eqref" href="#eq:NilpotencyOfQ">(2)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><munder><munder><mrow><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mphantom><mi>AA</mi></mphantom><mo>⇒</mo><mphantom><mi>AA</mi></mphantom><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> [Q,[Q,-] ] \;=\; [ \underset{= 0}{\underbrace{[Q,Q]}}, - ] - [ Q, [Q, -] ] \phantom{AA} \Rightarrow \phantom{AA} [ Q,[Q,-] ] = 0 </annotation></semantics></math></div> <p>and the <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>-property of the differential over the bracket follows again with the super Jacobi identity <a class="maruku-eqref" href="#eq:GradedJacobiIdentity">(1)</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mi>x</mi><mo stretchy="false">]</mo><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [Q,[x,y] ] \;=\; [ [Q,x],y] + (-1)^{deg(x)} [x, [Q,y] ] \,. </annotation></semantics></math></div> <h2 id="Examples">Examples</h2> <h3 id="basic_examples">Basic examples</h3> <p>Some obvious but important classes of examples are the following:</p> <div class="num_example" id="SuperVectorSpaceAsAbelianSuperLieAlgebra"> <h6 id="example">Example</h6> <p>every <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>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> <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> becomes a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> (def. <a class="maruku-ref" href="#SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces"></a>, prop. <a class="maruku-ref" href="#SuperLieAlgebraTraditional"></a>) by taking the super Lie bracket to be the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [-,-] = 0 \,. </annotation></semantics></math></div> <p>These may be called the “abelian” super Lie algebras.</p> </div> <div class="num_example" id="OrdinaryLieAlgebraAsSuperLieAlgebra"> <h6 id="example_2">Example</h6> <p>Every ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> becomes a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> (def. <a class="maruku-ref" href="#SuperLieAlgebraAsLieAlgebraInternalToSuperVectorSpaces"></a>, prop. <a class="maruku-ref" href="#SuperLieAlgebraTraditional"></a>) concentrated in even degrees. This constitutes a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LieAlg</mi><mo>↪</mo><mi>sLieAlg</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> LieAlg \hookrightarrow sLieAlg \,. </annotation></semantics></math></div> <p>which is a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> inclusion in that it has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LieAlg</mi><munderover><mo>⊥</mo><munder><mo>⟵</mo><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>⇝</mo></mover></munder><mo>↪</mo></munderover><mi>sLieAlg</mi></mrow><annotation encoding="application/x-tex"> LieAlg \underoverset {\underset{ \overset{ \rightsquigarrow}{(-)} }{\longleftarrow}} {\hookrightarrow} {\bot} sLieAlg </annotation></semantics></math></div> <p>given on the underlying super vector spaces by restriction to the even graded part</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>𝔰</mi><mo>⇝</mo></mover><mo>=</mo><msub><mi>𝔰</mi> <mi>even</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overset{\rightsquigarrow}{\mathfrak{s}} = \mathfrak{s}_{even} \,. </annotation></semantics></math></div></div> <h3 id="superpoincar_super_lie_algebras_supersymmetry">Super-Poincaré super Lie algebras (supersymmetry)</h3> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> and various of its polyvector extension are super-extension of the ordinary <a class="existingWikiWord" href="/nlab/show/Poincare+Lie+algebra">Poincare Lie algebra</a>. These are the <a class="existingWikiWord" href="/nlab/show/supersymmetry+algebras">supersymmetry algebras</a> in the strict original sense of the word. For more on this see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supersymmetry">geometry of physics – supersymmetry</a></em>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+q-Schur+algebra">super q-Schur algebra</a></p> </li> <li> <p>For every <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected topological space</a>, the <a class="existingWikiWord" href="/nlab/show/Whitehead+product">Whitehead product</a> makes its <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> into a super Lie algebra over the <a class="existingWikiWord" href="/nlab/show/ring">ring</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> – the <a href="Whitehead+product#SuperLieAlgebraStructure">Whitehead super Lie algebra</a>.</p> </li> <li> <p>higher super Lie algebras</p> <p>Just as <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> are <a class="existingWikiWord" href="/nlab/show/vertical+categorification">categorified</a> to <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</a>s and <a class="existingWikiWord" href="/nlab/show/L-infinity+algebroid">L-infinity algebroid</a>s, so super Lie algebras categorifie to <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a>s. A secretly famous example is the</p> <ul> <li><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></li> </ul> </li> </ul> <h3 id="SuperLieAlgebraInducedFromVectorSpace">Embedding tensors and tensor hierarchy</h3> <p>The following example is highlighted in <a href="tensor+hierarchy#Palmkvist13">Palmkvist 13, 3.1</a> (reviewed more clearly in <a href="tensor+hierarchy#LavauPalmkvist19">Lavau-Palmkvist 19, 2.4</a>) where it is attributed to I. L. Kantor (1970).</p> <div class="num_defn" id="TheSuperLieAlgebraInducedFromVectorSpace"> <h6 id="definition_5">Definition</h6> <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 <a class="existingWikiWord" href="/nlab/show/finite-dimensional+vector+space">finite-dimensional vector space</a> over some <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <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">k</annotation></semantics></math>.</p> <p>Define 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+vector+space">graded vector space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo>^</mo></mover><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Vect</mi> <mi>k</mi> <mi>ℤ</mi></msubsup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \widehat V \;\in \; Vect_k^{\mathbb{Z}} \,, </annotation></semantics></math></div> <p>concentrated in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\leq 1</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/recursion">recursively</a> as follows:</p> <p>For <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> we set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>V</mi><mo>^</mo></mover> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \widehat V_{1} \;\coloneqq\; V \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>0</mn><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \leq 0 \in \mathbb{Z}</annotation></semantics></math>, the component space in degree <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> is taken to be the vector space of <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> from <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> to the component space in degree <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \widehat V_{n-1} \;\coloneqq\; Hom_k( V, \widehat V_n ) \,. </annotation></semantics></math></div> <p>Hence:</p> <div class="maruku-equation" id="eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace"><span class="maruku-eq-number">(3)</span><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><msub><mover><mi>V</mi><mo>^</mo></mover> <mn>1</mn></msub></mtd> <mtd><mo>=</mo><mi>V</mi></mtd></mtr> <mtr><mtd><msub><mover><mi>V</mi><mo>^</mo></mover> <mn>0</mn></msub></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>=</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>⊗</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \widehat V_1 & = V \\ \widehat V_0 & = Hom_k(V,V) = \mathfrak{gl}(V) \\ \widehat V_{-1} & = Hom_k(V, Hom_k(V,V)) \simeq Hom_k(V \otimes V, V) \\ \widehat V_{-2} & = Hom_k(V, Hom_k(V, Hom_k(V,V))) \simeq Hom_k(V \otimes V \otimes V, V) \\ \vdots \end{aligned} </annotation></semantics></math></div> <p>Consider then the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of these component spaces as a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> with the <a class="existingWikiWord" href="/nlab/show/even+number">even number</a>/<a class="existingWikiWord" href="/nlab/show/odd+number">odd number</a>-degrees being in super-even/super-odd degree, respectively.</p> <p>On this <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> consider a <a class="existingWikiWord" href="/nlab/show/super+Lie+bracket">super Lie bracket</a> defined <a class="existingWikiWord" href="/nlab/show/recursion">recusively</a> as follows:</p> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mn>1</mn></msub><mo>=</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v_1, v_2 \in \widehat V_1 = V</annotation></semantics></math> we set</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><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [v_1, v_2] \;=\; 0 \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mi>n</mi><mo>≤</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f \in \widehat V_{n \leq 0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mn>1</mn></msub><mo>=</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \in \widehat V_1 = V</annotation></semantics></math> we set</p> <div class="maruku-equation" id="eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>v</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [f, v] \;\coloneqq\; f(v) </annotation></semantics></math></div> <p>Finally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f\in \widehat V_{ deg(f) \leq 0 }</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>≤</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">g\in \widehat V_{deg(g) \leq 0}</annotation></semantics></math> we set</p> <div class="maruku-equation" id="eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace"><span class="maruku-eq-number">(5)</span><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><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo></mtd> <mtd><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>v</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [f, g] & \colon\; v \;\mapsto\; [f, g(v)] - (-1)^{ deg(f) deg(g) } [ g, f(v) ] \\ \end{aligned} </annotation></semantics></math></div></div> <div class="num_remark" id="ConstraintsOnBracketInSuperLieAlgebraInducedByVectorSpace"> <h6 id="remark">Remark</h6> <p>By <a class="maruku-eqref" href="#eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace">(4)</a> the definition <a class="maruku-eqref" href="#eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace">(5)</a> is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo><mo>,</mo><mi>v</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>v</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>v</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [ [f,g],v ] \;=\; [f, [g,v] ] - (-1)^{ deg(f) deg(g) } [ g, [f,v] ] </annotation></semantics></math></div> <p>Hence <a class="maruku-eqref" href="#eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace">(5)</a> is already implied by <a class="maruku-eqref" href="#eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace">(4)</a> if the bracket is to satisfy the super Jacobi identity <a class="maruku-eqref" href="#eq:GradedJacobiIdentity">(1)</a>.</p> </div> <p>It remains to show that:</p> <div class="num_prop" id="SuperJacobiForSuperLieAlgebraInducedFromVectorSpace"> <h6 id="proposition_3">Proposition</h6> <p>Def. <a class="maruku-ref" href="#TheSuperLieAlgebraInducedFromVectorSpace"></a> indeed gives a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> in that the bracket <a class="maruku-eqref" href="#eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace">(5)</a> satisfies the super Jacobi identity <a class="maruku-eqref" href="#eq:GradedJacobiIdentity">(1)</a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We proceed by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>:</p> <p>By Remark <a class="maruku-ref" href="#ConstraintsOnBracketInSuperLieAlgebraInducedByVectorSpace"></a> we have that the super Jacobi identity holds for for all <a class="existingWikiWord" href="/nlab/show/triples">triples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>∈</mo><mover><mi>V</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">f_1, f_2, f_3 \in \widehat{V}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">deg(f_3) \geq 0</annotation></semantics></math>.</p> <p>Now assume that the super Jacobi identity has been shown for triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_1, f_2, f_3(v))</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>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_1, f_3, f_2(v))</annotation></semantics></math>, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">v \in V</annotation></semantics></math>. The following computation shows that then it holds for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_1, f_2, f_3)</annotation></semantics></math>:</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><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mstyle mathcolor="green"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mstyle mathcolor="orange"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mstyle mathcolor="blue"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mstyle mathcolor="cyan"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><munder><mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mstyle mathcolor="green"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle><mo>−</mo><mstyle mathcolor="blue"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><munder><mrow><mstyle mathcolor="orange"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle><mo>−</mo><mstyle mathcolor="cyan"><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mstyle></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>f</mi> <mn>3</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>,</mo><mo stretchy="false">[</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [f_1, [f_2, f_3] ] (v) & = [ f_1, [f_2, f_3](v) ] - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ [ f_2, f_3 ], f_1(v) ] \\ & = [ f_1, [ f_2, f_3(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_2)deg(f_3)} [ f_1, [ f_3, f_2(v) ] ] \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} [ f_2, [ f_3, f_1(v) ] ] \\ & \phantom{=} + (-1)^{deg(f_1)(deg(f_2) + deg(f_3)) + deg(f_2)deg(f_3)} [ f_3, [ f_2, f_1(v) ] ] \\ & = [ f_1, [ f_2, f_3(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{green} [ f_2, [ f_1, f_3(v) ] ] } \\ & \phantom{=} - (-1)^{deg(f_2) deg(f_3)} \big( [ f_1, [ f_3, f_2(v) ] ] - (-1)^{deg(f_1) deg(f_3)} { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } \big) \\ & \phantom{=} - (-1)^{deg(f_1)(deg(f_2) + deg(f_3))} \big( [ f_2, [ f_3, f_1(v) ] ] - (-1)^{deg(f_1)deg(f_3)} { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2 ) + deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( [ f_3, [ f_2, f_1(v) ] ] - (-1)^{deg(f_1) deg(f_2)} { \color{cyan} [ f_3, [ f_1, f_2(c) ] ] } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_2)} \big( \underset{ = 0 }{ \underbrace{ + { \color{green} [ f_2, [ f_1, f_3(v) ] ] } - { \color{blue} [ f_2, [ f_1, f_3(v) ] ] } } } \big) \\ & \phantom{=} + (-1)^{deg(f_1) deg(f_3) + deg(f_2) deg(f_3)} \big( \underset{ = 0 }{ \underbrace{ { \color{orange} [ f_3, [ f_1, f_2(v) ] ] } - { \color{cyan} [ f_3, [ f_1, f_2(v) ] ] } } } \big) \\ & = \big[ [f_1, f_2], f_3(v) \big] \\ & \phantom{=} - (-1)^{ deg(f_2) deg(f_3) } \big[ [f_1, f_3], f_2(c) \big] \\ & \phantom{=} + (-1)^{ deg(f_1) deg(f_2) } \big[ f_2, [f_1, f_3](v) \big] \\ & \phantom{=} - (-1)^{ deg(f_3)( deg(f_1) + deg(f_2) ) } \big[ f_3, [f_1, f_2](v) \big] \\ & = \big[ [f_1, f_2], f_3 \big](v) + (-1)^{deg(f_1)deg(f_2))} \big[ f_2, [f_1, f_3] \big](v) \end{aligned} </annotation></semantics></math></div> <blockquote> <p>(Fine, but is this sufficient to induct over the full range of all three degrees?)</p> </blockquote> </div> <div class="num_example" id="LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mn>0</mn></msub><mo>=</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in \widehat V_0 = Hom_k(V,V)</annotation></semantics></math> <a class="maruku-eqref" href="#eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace">(3)</a> we have that the bracket on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat V</annotation></semantics></math> in Def. <a class="maruku-ref" href="#SuperLieAlgebraInducedFromVectorSpace"></a> restricts to</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>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>g</mi><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mi>g</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [f,g](v) \;=\; [f,g(v)] - [g,f(v)] \;=\; f(g(v)) - g(f(v)) </annotation></semantics></math></div> <p>(by combining <a class="maruku-eqref" href="#eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace">(5)</a> with <a class="maruku-eqref" href="#eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace">(4)</a>).</p> <p>This is the <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> of the <a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{gl}(V)</annotation></semantics></math>, as indicated on the right in <a class="maruku-eqref" href="#eq:ExplicitComponentSpacesOfSuperLieAlgebraInducedFromVectorSpace">(3)</a>.</p> </div> <div class="num_prop" id="EmbeddingTensorViaSuperLieAlgebraInducedFromVectorSpace"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/embedding+tensors">embedding tensors</a> are square-0 elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{V}</annotation></semantics></math>)</strong></p> <p>Let <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">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>.</p> <p>An element in degree -1 of the <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat V</annotation></semantics></math> from Def. <a class="maruku-ref" href="#SuperLieAlgebraInducedFromVectorSpace"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo>∈</mo><msub><mover><mi>V</mi><mo>^</mo></mover> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>≃</mo><msub><mi>Hom</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Theta \in \widehat V_{-1} \simeq Hom_{k}(V, \mathfrak{gl}(V)) \,, </annotation></semantics></math></div> <p>which by Example <a class="maruku-ref" href="#LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace"></a> is identified with a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>⟶</mo><mi>𝔤</mi><mo>≔</mo><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Theta \;\colon\; V \longrightarrow \mathfrak{g} \coloneqq \mathfrak{gl}(V) </annotation></semantics></math></div> <p>from <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> to the <a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a> on <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>, is square-0 <a class="maruku-eqref" href="#eq:NilpotencyOfQ">(2)</a> precisely if it is an <em><a class="existingWikiWord" href="/nlab/show/embedding+tensor">embedding tensor</a></em>, in that:</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>Θ</mi><mo>,</mo><mi>Θ</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mphantom><mi>AAA</mi></mphantom><mo>⇔</mo><mphantom><mi>AAA</mi></mphantom><mo stretchy="false">[</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Theta, \Theta] \;=\; 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} [\Theta(v_1), \Theta(v_2) ] \;=\; \Theta( \rho_{\Theta(v)1)}(v_2) ) \,. </annotation></semantics></math></div> <p>Here on the right, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{gl}(V)</annotation></semantics></math>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> denotes the canonical <a class="existingWikiWord" href="/nlab/show/Lie+algebra+action">Lie algebra action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{gl}(V)</annotation></semantics></math> on <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>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By unwinding of the definition <a class="maruku-eqref" href="#eq:SuperLieBracketOnDegree0InSuperLieAlgebraInducedFromVectorSpace">(4)</a> and <a class="maruku-eqref" href="#eq:SuperLieBracketInSuperLieAlgebraInducedFromVectorSpace">(5)</a> and using again Example <a class="maruku-ref" href="#LieBracketOnglInsideSuperLieAlgebraInducedFromVectorSpace"></a> we compute as follows:</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><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">[</mo><mi>Q</mi><mo>,</mo><munder><munder><mrow><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mpadded width="0" lspace="-50%width"><mrow><mo>=</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></munder><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>Q</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Q</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>ρ</mi> <mrow><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mo stretchy="false">[</mo><mi>Θ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Θ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \big( \tfrac{1}{2} [Q,Q](v_1) \big)(v_2) & = [Q, Q(v_1)](v_2) \\ & = [Q, \underset{ \mathclap{ = \rho_{\Theta(v_1)}(v_2) } } { \underbrace{ (Q(v_1))(v_2) } } ] - [Q(v_1), Q(v_2)] \\ & = \Theta( \rho_{\Theta(v_1)}(v_2) ) - [ \Theta(v_1), \Theta(v) ] \end{aligned} </annotation></semantics></math></div></div> <div class="num_prop" id="EmbeddingTensorsInduceTensorHierarchies"> <h6 id="remark_2">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/embedding+tensors">embedding tensors</a> induce <a class="existingWikiWord" href="/nlab/show/tensor+hierarchies">tensor hierarchies</a>)</strong></p> <p>In view of the relation between <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> and <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> (<a href="#RelationToDGLieAlgebras">above</a>), Prop. <a class="maruku-ref" href="#EmbeddingTensorViaSuperLieAlgebraInducedFromVectorSpace"></a> says that every choice of an <a class="existingWikiWord" href="/nlab/show/embedding+tensor">embedding tensor</a> for a <a class="existingWikiWord" href="/nlab/show/faithful+representation">faithful representation</a> on 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>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> induces a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>V</mi><mo>^</mo></mover><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>,</mo><mo>∂</mo><mo>≔</mo><mo stretchy="false">[</mo><mi>Θ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\widehat V, [-,-], \partial \coloneqq [\Theta, -])</annotation></semantics></math>.</p> <p>According to <a href="tensor+hierarchy#Palmkvist13">Palmkvist 13, 3.1</a>, <a href="tensor+hierarchy#LavauPalmkvist19">Lavau-Palmkvist 19, 2.4</a> this <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> (or some extension of some sub-algebra of it) is the <em><a class="existingWikiWord" href="/nlab/show/tensor+hierarchy">tensor hierarchy</a></em> associated with the embedding tensor.</p> </div> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%E2%80%93%C5%81opusza%C5%84ski%E2%80%93Sohnius+theorem">Haag–Łopuszański–Sohnius theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supersymmetry">geometry of physics – supersymmetry</a></p> </li> </ul> <h2 id="references">References</h2> <p>According to <a href="#Kac77b">V. Kac 1977b</a> the definition of super Lie algebra is originally due to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Felix+Berezin">Felix Berezin</a>, G. I. Kac, Math. Sbornik <strong>82</strong> (1970) 343-351 <blockquote> <p>(in Russian)</p> </blockquote> </li> </ul> <p>See also:</p> <ul> <li id="Kantor70"> <p><a class="existingWikiWord" href="/nlab/show/Isaiah+Kantor">Isaiah Kantor</a>, <em>Graded Lie algebras</em>, Trudy Sem. Vektor. Tenzor. Anal 15 (1970): 227-266.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Felix+A.+Berezin">Felix A. Berezin</a> (edited by <a class="existingWikiWord" href="/nlab/show/Alexandre+A.+Kirillov">Alexandre A. Kirillov</a>): <em>Lie Superalgebras</em>, chapters I.5 and II.1 in: <em>Introduction to Superanalysis</em>, Mathematical Physics and Applied Mathematics <strong>9</strong>, Springer (1987) [<a href="https://doi.org/10.1007/978-94-017-1963-6_6">doi:10.1007/978-94-017-1963-6_6</a>, <a href="https://doi.org/10.1007/978-94-017-1963-6_7">doi:10.1007/978-94-017-1963-6_7</a>]</p> </li> </ul> <p>The original references on the classification of super Lie algebras:</p> <ul> <li id="Kac77a"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Kac">Victor Kac</a>, <em>Lie superalgebras</em>, Advances in Math. <strong>26</strong> 1 (1977) 8-96 [<a href="https://doi.org/10.1016/0001-8708(77)90017-2">doi:10.1016/0001-8708(77)90017-2</a>]</p> </li> <li id="Kac77b"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Kac">Victor Kac</a>: <em>A sketch of Lie superalgebra theory</em>, Comm. Math. Phys. <strong>53</strong> 1 (1977) 31-64 [<a href="https://projecteuclid.org/euclid.cmp/1103900590">euclid:1103900590</a>, <a href="https://doi.org/10.1007/BF01609166">doi:10.1007/BF01609166</a>]</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Werner+Nahm">Werner Nahm</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Rittenberg">Vladimir Rittenberg</a>, <a class="existingWikiWord" href="/nlab/show/Manfred+Scheunert">Manfred Scheunert</a>, <em>The classification of graded Lie algebras</em>, Physics Letters B <strong>61</strong> 4 (1976) 383-384 [<a href="https://doi.org/10.1016/0370-2693(76)90594-3">doi:10.1016/0370-2693(76)90594-3</a>]</p> </li> <li> <p>M. Parker, <em>Classification Of Real Simple Lie Superalgebras Of Classical Type</em>, J.Math.Phys. 21 (1980) 689-697 (<a href="http://inspirehep.net/record/157627?ln=en">spire</a>)</p> </li> </ul> <p>Further discussion specifically of <a href="#supersymmetry#Classification">classification of supersymmetry</a>:</p> <ul> <li id="Nahm78"> <p><a class="existingWikiWord" href="/nlab/show/Werner+Nahm">Werner Nahm</a>, <em><a class="existingWikiWord" href="/nlab/show/Supersymmetries+and+their+Representations">Supersymmetries and their Representations</a></em>, Nucl. Phys. B <strong>135</strong> (1978) 149 [<a href="https://inspirehep.net/record/120988/">spire</a>, <a href="http://cds.cern.ch/record/132743/files/197709213.pdf">pdf</a>]</p> </li> <li id="Shnider88"> <p><a class="existingWikiWord" href="/nlab/show/Steven+Shnider">Steven Shnider</a>, <em>The superconformal algebra in higher dimensions</em>, Letters in Mathematical Physics <strong>16</strong> 4 (1988) 377-383 [<a href="https://doi.org/10.1007/BF00402046">doi:10.1007/BF00402046</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Kac">Victor Kac</a>, <em>Classification of supersymmetries</em>, Proceedings of the ICM, Beijing 2002, vol. 1, 319–344 (<a href="http://arxiv.org/abs/math-ph/0302016">arXiv:math-ph/0302016</a>)</p> </li> </ul> <p>Introductions and surveys:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Rittenberg">Vladimir Rittenberg</a>: <em>A guide to Lie superalgebras</em>, in P. Kramers, A. Rieckers (eds.): <em>Group Theoretical Methods in Physics</em>, Lecture Notes in Physics <strong>79</strong> Springer (1978) 3-21 [<a href="https://doi.org/10.1007/3-540-08848-2_1">doi:10.1007/3-540-08848-2_1</a>, <a class="existingWikiWord" href="/nlab/files/Rittenberg-GuideToLieSuperalgebras.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Manfred+Scheunert">Manfred Scheunert</a>, <em>The theory of Lie superalgebras. An introduction</em>, Lect. Notes Math. <strong>716</strong> (1979) [<a href="https://doi.org/10.1007/BFb0070929">doi:10.1007/BFb0070929</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dimitry+A.+Leites">Dimitry A. Leites</a>, <em>Lie Superalgebras</em>, J. Soviet Math. <strong>30</strong> (1985) 2481-2512 [<a href="http://dx.doi.org/10.1007/BF02249121">doi:10.1007/BF02249121</a>]</p> </li> <li id="CastellaniDAuriaFre"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <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>, section II.2 of: <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991) [<a href="https://doi.org/10.1142/0224">doi:10.1142/0224</a>, toc: <a class="existingWikiWord" href="/nlab/files/CDF91-TOC.pdf" title="pdf">pdf</a>, chII.2: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChII2.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(with an eye towards <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>)</p> </blockquote> </li> <li id="Farmer84"> <p>Richard J. Farmer, <em>Orthosymplectic superalgebras in mathematics and science</em>, PhD Thesis (1984) [<a href="http://eprints.utas.edu.au/19542">eprints:19542</a>, <a class="existingWikiWord" href="/nlab/files/Farmer-OSpSuperalgebra.pdf" title="pdf">pdf</a>]</p> </li> <li> <p>Luc Frappat, Antonio Sciarrino, Paul Sorba: <em>Structure of basic Lie superalgebras and of their affine extensions</em>, Comm. Math. Phys. <strong>121</strong> 3 (1989) 457-500 [<a href="https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-121/issue-3/Structure-of-basic-Lie-superalgebras-and-of-their-affine-extensions/cmp/1104178142.full">euclid:cmp/1104178142</a>]</p> </li> <li> <p>Luc Frappat, Antonio Sciarrino, Paul Sorba: <em>Dictionary on Lie Superalgebras</em>, Academic Press (2000) [<a href="http://arxiv.org/abs/hep-th/9607161">arXiv:hep-th/9607161</a>, ISBN:978-0122653407]</p> </li> <li> <p>Groeger, <em>Super Lie groups and super Lie algebras</em>, lecture notes 2011 (<a href="http://www.mathematik.hu-berlin.de/~groegerj/teaching_files/lecture12.pdf">pdf</a>)</p> </li> <li> <p>D. Westra, <em>Superrings and supergroups</em> (<a href="http://www.mat.univie.ac.at/~michor/westra_diss.pdf">pdf</a>)</p> </li> <li id="Minwalla98"> <p><a class="existingWikiWord" href="/nlab/show/Shiraz+Minwalla">Shiraz Minwalla</a>, <em>Restrictions imposed by superconformal invariance on quantum field theories</em>, Adv. Theor. Math. Phys. <strong>2</strong> (1998) 781 [<a href="http://arxiv.org/abs/hep-th/9712074">arXiv:hep-th/9712074</a>]</p> </li> </ul> <p>Discussion in the <a class="existingWikiWord" href="/nlab/show/presheaf+topos">presheaf topos</a> over <a class="existingWikiWord" href="/nlab/show/superpoints">superpoints</a>:</p> <ul> <li id="Sachse08"><a class="existingWikiWord" href="/nlab/show/Christoph+Sachse">Christoph Sachse</a>, Section 3 of: <em>A Categorical Formulation of Superalgebra and Supergeometry</em> [<a href="http://arxiv.org/abs/0802.4067">arXiv:0802.4067</a>]</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a> for super Lie algebras includes</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, <em>Extensions of super Lie algebras</em>, J. Lie Theory <strong>15</strong> 1 (2005) 125-134 [<a href="http://arxiv.org/abs/math/0101190">arXiv:math/0101190</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Lie+algebra+weight+systems">Lie algebra weight systems</a> arising from <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a>:</p> <ul> <li id="FFKV97"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%E2%80%99Farrill">José Figueroa-O’Farrill</a>, <a class="existingWikiWord" href="/nlab/show/Takashi+Kimura">Takashi Kimura</a>, <a class="existingWikiWord" href="/nlab/show/Arkady+Vaintrob">Arkady Vaintrob</a>, <em>The universal Vassiliev invariant for the Lie superalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤𝔩</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{gl}(1\vert1)</annotation></semantics></math></em>, Commun. Math. Phys. 185 (1997) 93-127 (<a href="https://arxiv.org/abs/q-alg/9602014">arXiv:q-alg/9602014</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> (see also the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">brane scan</a>) in relation to <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integrable forms</a> of coset <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Roberto+Catenacci">Roberto Catenacci</a>, <a class="existingWikiWord" href="/nlab/show/C.+A.+Cremonini">C. A. Cremonini</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Antonio+Grassi">Pietro Antonio Grassi</a>, <a class="existingWikiWord" href="/nlab/show/Simone+Noja">Simone Noja</a>, <em>Cohomology of Lie Superalgebras: Forms, Integral Forms and Coset Superspaces</em> (<a href="https://arxiv.org/abs/2012.05246">arXiv:2012.05246</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 10, 2024 at 08:48:53. 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