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href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> <h4 id="supergeometry">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> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#LieAlgebraCohomology'>Lie algebra cohomology</a></li> <li><a href='#extensions'>Extensions</a></li> <ul> <li><a href='#super_algebra_extensions'>Super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra extensions</a></li> <li><a href='#PolyvectorExtensions'>Extended super Poincaré Lie algebra – Polyvector extensions</a></li> <ul> <li><a href='#as_current_algebras_of_the_gs_super_branes'>As current algebras of the GS super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes</a></li> <li><a href='#PolyvectorExtensionsAsAutomorphismLieAlgebras'>As automorphism Lie algebras of Lie <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>-superalgebras</a></li> </ul> </ul> <li><a href='#Contractions'>Contractions</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#PolyvectorExtensionRefs'>Polyvector extensions</a></li> <li><a href='#ReferencesLieAlgebraCohomology'>Super Lie algebra cohomology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>super Poincaré Lie algebra</em> is a <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a> extension of a <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> is the <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> (except for the signature of the metric).</p> <h2 id="Definition">Definition</h2> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math> and consider <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R} ^{d-1,1}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(d-1,1)</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S \in Rep(Spin(d-1,1)) </annotation></semantics></math></div> <p>be a real <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a> (<a class="existingWikiWord" href="/nlab/show/Majorana+spinor">Majorana spinor</a>), which has the property (<a href="Majorana+spinor#SpinorToVectorBilinearPairing">def.</a>, <a href="Majorana+spinor#SpinorToVectorPairing">prop.</a>) that there exists 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>Γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>S</mi><mo>⊗</mo><mi>S</mi><mo>⟶</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \Gamma \;\colon\; S \otimes S \longrightarrow \mathbb{R}^{d-1,1} </annotation></semantics></math></div> <p>which is</p> <ol> <li> <p>symmetric:</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(V)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/equivariance">equivariant</a> (a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/spin+representations">spin representations</a>).</p> </li> </ol> <p>For a classification of spin representations with this property see at <em><a class="existingWikiWord" href="/nlab/show/spin+representations">spin representations</a></em> the sections <em><a href="spin+representation#RealIrreducibleSpinRepresentationInLorentzSignature">real irreducible spin representations in Lorentz signature</a></em> and <em><a href="spin+representation#SuperPoincareBrackets">super Poincaré brackets</a></em>. For explicit construction in components see at <em><a class="existingWikiWord" href="/nlab/show/Majorana+spinor">Majorana spinor</a></em> the section <em><a href="Majorana+spinor#TheSpinorPairingToVectors">The spinor pairing to vectors</a></em>.</p> <div class="num_defn" id="SuperPoincare"> <h6 id="definition_2">Definition</h6> <p>The <strong>super Poincaré Lie algebra</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔰𝔦𝔰𝔬</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}_S(d-1,1)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> with respect to the <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with symmetric and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(V)</annotation></semantics></math>-equivariant pairing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>⊗</mo><mi>S</mi><mo>→</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Gamma \colon S \otimes S \to \mathbb{R}^{d-1,1}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">Lie algebra extension</a> of the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+Lie+algebra">Poincaré Lie algebra</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">\Pi S</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> taken in odd degree)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mi>S</mi><mo>⟶</mo><msub><mi>𝔰𝔦𝔰𝔬</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>⋉</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Pi S \longrightarrow \mathfrak{siso}_S(d-1,1) \longrightarrow \mathfrak{iso}(d-1,1) \simeq \mathbb{R}^{d-1,1} \ltimes \mathfrak{so}(d-1,1) \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(d-1,1)</annotation></semantics></math> with those in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is the given <a class="existingWikiWord" href="/nlab/show/action">action</a>, the Lie bracket of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math> with those on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is trivial, and the Lie bracket of two elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_1, s_2 \in S</annotation></semantics></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</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 stretchy="false">[</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>≔</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [s_1,s_2] \coloneqq \Gamma(s_1,s_2) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>It is precisely the symmetry and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(V)</annotation></semantics></math>-equivariant assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> that makes this a well defined <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>: the symmetry corresponds to the graded skew-symmetry of the <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> on elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, which are regarded as odd, and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(V)</annotation></semantics></math>-equivariance yields the nontrivial <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>o</mi><mo>∈</mo><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">o \in \mathfrak{so}(d-1,1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s_1, s_2 \in S</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>o</mi><mo>,</mo><msub><mi>s</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">[</mo><mi>o</mi><mo>,</mo><msub><mi>s</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>o</mi><mo>,</mo><mi>Γ</mi><mo stretchy="false">(</mo><msub><mi>s</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>s</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"> \Gamma([o,s_1], s_2) + \Gamma(s_1, [o,s_2]) = [o, \Gamma(s_1,s_2)] \,. </annotation></semantics></math></div></div> <div class="num_remark" id="CEAlgebraOfSuperPoincare"> <h6 id="remark_2">Remark</h6> <p>By the general discussion at <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, we may characterize the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔰𝔦𝔰𝔬</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}_S(D-1,1)</annotation></semantics></math> by its CE super-<a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-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><msub><mi>𝔰𝔦𝔰𝔬</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{siso}_S(D-1,1))</annotation></semantics></math> “of <a class="existingWikiWord" href="/nlab/show/left-invariant+1-forms">left-invariant 1-forms</a>” on its group manifold.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ω</mi> <mi>a</mi></msub><msup><mrow></mrow> <mi>b</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{\omega_a{}^b\}_{a,b}</annotation></semantics></math> for the canonical basis of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal</a> <a class="existingWikiWord" href="/nlab/show/matrix+Lie+algebra">matrix Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>D</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(D-1,1)</annotation></semantics></math> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>ψ</mi> <mi>α</mi></msub><msub><mo stretchy="false">}</mo> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\{\psi_\alpha\}_\alpha</annotation></semantics></math> for a corresponding <a class="existingWikiWord" href="/nlab/show/basis">basis</a> of the <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><msub><mi>𝔰𝔦𝔰𝔬</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{siso}_N(d-1,1))</annotation></semantics></math> is generated on</p> <ul> <li> <p>elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>e</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e^a\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\omega^{ a b}\}</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>even</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,even)</annotation></semantics></math></p> </li> <li> <p>and elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ψ</mi> <mi>α</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\psi^\alpha\}</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>odd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1,odd)</annotation></semantics></math></p> </li> </ul> <p>with the differential defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>=</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mrow><mi>b</mi><mi>c</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> d_{CE} \omega^{a b} = \omega^a{}_b \wedge \omega^{b c} </annotation></semantics></math></div><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>CE</mi></msub><msup><mi>e</mi> <mi>a</mi></msup><mo>=</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>e</mi> <mi>b</mi></msup><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>ψ</mi><mo stretchy="false">¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi></mrow><annotation encoding="application/x-tex"> d_{CE} e^{a } = \omega^a{}_b \wedge e^b + \frac{i}{2}\bar \psi \Gamma^a \psi </annotation></semantics></math></div><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>CE</mi></msub><mi>ψ</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>∧</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mi>ψ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{CE} \psi = \frac{1}{4} \omega^{ a b} \wedge \Gamma_{a b} \psi \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>Removing all terms involving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> here yields the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of the <a class="existingWikiWord" href="/nlab/show/super+translation+algebra">super translation algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>D</mi><mo>;</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{D;N}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The abstract generators in def. <a class="maruku-ref" href="#CEAlgebraOfSuperPoincare"></a> are identified with <a class="existingWikiWord" href="/nlab/show/left+invariant+1-forms">left invariant 1-forms</a> on the <a class="existingWikiWord" href="/nlab/show/super-translation+group">super-translation group</a> as follows.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>θ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x^a, \theta^\alpha)</annotation></semantics></math> be the canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> on the <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d|N}</annotation></semantics></math> underlying the super translation group. Then the identification is</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ψ</mi> <mi>α</mi></msup><mo>=</mo><mi>d</mi><msup><mi>θ</mi> <mi>α</mi></msup></mrow><annotation encoding="application/x-tex">\psi^\alpha = d \theta^\alpha</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>e</mi> <mi>a</mi></msup><mo>=</mo><mi>d</mi><msup><mi>x</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>d</mi><mi>θ</mi></mrow><annotation encoding="application/x-tex">e^a = d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta</annotation></semantics></math>.</p> </li> </ul> <p>This then gives the formula for the differential of the super-<a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> in def. <a class="maruku-ref" href="#CEAlgebraOfSuperPoincare"></a> as</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><mi>d</mi><msup><mi>e</mi> <mi>a</mi></msup></mtd> <mtd><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mi>x</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>d</mi><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mi>d</mi><mover><mi>θ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>d</mi><mi>θ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mfrac><mi>i</mi><mn>2</mn></mfrac><mover><mi>ψ</mi><mo>¯</mo></mover><msup><mi>Γ</mi> <mi>a</mi></msup><mi>ψ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d e^a & = d (d x^a + \frac{i}{2} \overline{\theta} \Gamma^a d \theta) \\ & = \frac{i}{2} d \overline{\theta}\Gamma^a d \theta \\ & = \frac{i}{2} \overline{\psi}\Gamma^a \psi \end{aligned} \,. </annotation></semantics></math></div></div> <h2 id="properties">Properties</h2> <h3 id="LieAlgebraCohomology">Lie algebra cohomology</h3> <p>The super Poincaré Lie algebra has, on top of the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cocycles</a> that it inherits from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(n)</annotation></semantics></math>, a discrete number of exceptiona cocycles bilinear in the spinors, on the <a class="existingWikiWord" href="/nlab/show/super+translation+algebra">super translation algebra</a>, that exist only in very special dimensions.</p> <p>The following theorem has been stated at various placed in the physics literature (known there as the <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>-symmetry in <em><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functionals">Green-Schwarz action functionals</a></em> for super-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/branes">branes</a> on <a class="existingWikiWord" href="/nlab/show/super-Minkowski+spacetime">super-Minkowski spacetime</a>). A full proof is in <a href="#Brandt12-13">Brandt 12-13</a>. The following uses the notation in terms of <a class="existingWikiWord" href="/nlab/show/division+algebras">division algebras</a> (<a href="#BaezHuerta10">Baez-Huerta 10</a>).</p> <p><strong>Theorem</strong></p> <ul> <li> <p>In dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>10</mn></mrow><annotation encoding="application/x-tex">d = 3,4,6, 10</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}(d-1,1)</annotation></semantics></math> has a nontrivial 3-cocycle given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ψ</mi><mo>,</mo><mi>ϕ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>g</mi><mo stretchy="false">(</mo><mi>ψ</mi><mo>⋅</mo><mi>ϕ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\psi, \phi, A) \mapsto g(\psi \cdot \phi, A) </annotation></semantics></math></div> <p>for spinors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>,</mo><mi>ϕ</mi><mo>∈</mo><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">\psi, \phi \in \mathcal{S}</annotation></semantics></math> and vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{T}</annotation></semantics></math>, and 0 otherwise.</p> </li> <li> <p>In dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">d = 4,5,7, 11</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}(d-1,1)</annotation></semantics></math> has a nontrivial 4-cocycle given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ψ</mi><mo>,</mo><mi>Φ</mi><mo>,</mo><mi>𝒜</mi><mo>,</mo><mi>ℬ</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">⟨</mo><mi>Ψ</mi><mo>,</mo><mo stretchy="false">(</mo><mi>𝒜</mi><mi>ℬ</mi><mo>−</mo><mi>ℬ</mi><mi>𝒜</mi><mo stretchy="false">)</mo><mi>Φ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> (\Psi, \Phi, \mathcal{A}, \mathcal{B}) \mapsto \langle \Psi , (\mathcal{A}\mathcal{B}- \mathcal{B} \mathcal{A})\Phi \rangle </annotation></semantics></math></div> <p>for spinors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>,</mo><mi>Φ</mi><mo>∈</mo><mi>𝒮</mi></mrow><annotation encoding="application/x-tex">\Psi, \Phi \in \mathcal{S}</annotation></semantics></math> and vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>,</mo><mi>ℬ</mi><mo>∈</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}, \mathcal{B} \in \mathcal{V}</annotation></semantics></math>, with the commutator taken in the Clifford algebra.</p> </li> </ul> <p>The 4-cocycle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">d = 11</annotation></semantics></math> is the one that induces the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>.</p> <p>All these cocycles are controled by the relevant <a class="existingWikiWord" href="/nlab/show/Fierz+identities">Fierz identities</a>.</p> <h3 id="extensions">Extensions</h3> <h4 id="super_algebra_extensions">Super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra extensions</h4> <p>The <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a> <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">infinity-Lie algebra cohomology</a> of the super Poincaré Lie algebra corresponding to the <a href="#LieAlgebraCohomology">above</a> cocycles involves</p> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <strong>super-Poincaré Lie algebra</strong></p> <h4 id="PolyvectorExtensions">Extended super Poincaré Lie algebra – Polyvector extensions</h4> <p>The super-Poincaré Lie algebra has a class of super <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a> called <em><a class="existingWikiWord" href="/nlab/show/extended+supersymmetry">extended supersymmetry</a></em> algebras or <em>polyvector extensions</em> , because they involve additional generators that transforn as skew-symmetric <a class="existingWikiWord" href="/nlab/show/tensors">tensors</a>. A complete classification is in (<a href="#ACDP">ACDP</a>).</p> <p>For instance the “<a class="existingWikiWord" href="/nlab/show/M-theory+Lie+algebra">M-theory Lie algebra</a>” is a polyvector extension of the super Poincaré Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔰𝔦𝔰𝔬</mi> <mrow><mi>N</mi><mo>=</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{siso}_{N=1}(10,1)</annotation></semantics></math> by polyvectors of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p = 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>=</mo><mn>5</mn></mrow><annotation encoding="application/x-tex">p=5</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> and the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> in the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a>), see below <a href="#PolyvectorExtensionsAsAutomorphismLieAlgebras">Polyvector extensions as automorphism Lie algebras</a>.</p> <h5 id="as_current_algebras_of_the_gs_super_branes">As current algebras of the GS super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-branes</h5> <p>The polyvector extensions arise as the super Lie algebras of <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+super+p-brane+sigma-models">Green-Schwarz super p-brane sigma-models</a> (<a href="#AGIT89">AGIT 89</a>).</p> <h5 id="PolyvectorExtensionsAsAutomorphismLieAlgebras">As automorphism Lie algebras of Lie <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>-superalgebras</h5> <p>At least some of the <a href="#PolyvectorExtensions">polyvector extensions</a> of the super Poincaré Lie algebra arise as the <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> super Lie algebras of the <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a> <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">extensions</a> classified by the cocycles discussed above.</p> <p>For instance the automorphisms of the <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a> gives the “<a class="existingWikiWord" href="/nlab/show/M-theory+Lie+algebra">M-theory Lie algebra</a>”-extension of super-Poincaré in 11-dimensions (<a href="#FSS13">FSS 13</a>). This is also discussed at <em><a href="supergravity+Lie+3-algebra#Polyvector">supergravity Lie 3-algebra – Polyvector extensions</a></em>.</p> <h3 id="Contractions">Contractions</h3> <p>One may consider <a class="existingWikiWord" href="/nlab/show/symmetry+breaking">breaking</a> super-Poincaré invariance by passage to non-relativistic or ultrarelativistic limits, formally understood as taking the <a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a> to <a class="existingWikiWord" href="/nlab/show/infinity"><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></a> or to <a class="existingWikiWord" href="/nlab/show/zero">0</a>, respectively, referred to as <em>Galilean</em> of <em>Carrollian</em> <a class="existingWikiWord" href="/nlab/show/limit+of+a+sequence">limits</a>, respectively. This is achieved via the process of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+contraction">contraction</a>, as described in <a href="#IW53">İnönü & Wigner 1953</a>.</p> <p>The corresponding super Lie algebras are defined as follows (see e.g. Section 2 of <a href="#KN23">Koutrolikos & Najafizadeh 2023</a>). We denote the generators of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>d</mi></mrow><annotation encoding="application/x-tex">4d</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>1</mn></mrow><annotation encoding="application/x-tex">N=1</annotation></semantics></math> super-Poincaré algebra as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>J</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow></msub><mo>,</mo><msub><mi>P</mi> <mi>μ</mi></msub><mo>,</mo><msub><mi>Q</mi> <mi>α</mi></msub><mo>,</mo><msub><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{ J_{\mu\nu}, P_{\mu}, Q_{\alpha}, \bar{Q}_{\dot \alpha} \}</annotation></semantics></math>.</p> <p>The <strong>super-Carroll</strong> algebra is defined by the rescaled generators</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.8em" minsize="1.8em">{</mo><msubsup><mi>K</mi> <mi>i</mi> <mi>C</mi></msubsup><mo>≔</mo><mi>c</mi><msub><mi>J</mi> <mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>,</mo><msubsup><mi>P</mi> <mn>0</mn> <mi>C</mi></msubsup><mo>≔</mo><mi>c</mi><msub><mi>P</mi> <mn>0</mn></msub><mo>,</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>C</mi></msubsup><mo>≔</mo><msub><mi>J</mi> <mi>ij</mi></msub><mo>,</mo><msubsup><mi>P</mi> <mi>i</mi> <mi>C</mi></msubsup><mo>≔</mo><msub><mi>P</mi> <mi>i</mi></msub><mo>,</mo><msubsup><mi>Q</mi> <mi>α</mi> <mi>C</mi></msubsup><mo>≔</mo><msqrt><mi>c</mi></msqrt><msub><mi>Q</mi> <mi>α</mi></msub><mo>,</mo><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover> <mi>C</mi></msubsup><mo>≔</mo><msub><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover></msub><mo maxsize="1.8em" minsize="1.8em">}</mo></mrow><annotation encoding="application/x-tex"> \Big\{K^C _i \coloneqq c J_{0i} , P^C _0 \coloneqq c P_0, J_{ij} ^C \coloneqq J_{ij} , P_{i} ^C \coloneqq P_{i}, Q^C _{\alpha} \coloneqq \sqrt{c} Q_{\alpha} , \bar{Q}^C _{\dot \alpha} \coloneqq \bar{Q} _{\dot \alpha} \Big\} </annotation></semantics></math></div> <p>satisfying the commutation relations inherited from the super-Poincaré algebra, with the exceptions that now the commutators</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left center left"><mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>K</mi> <mi>i</mi> <mrow><msup><mo></mo><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">rm</mo><mi>C</mi><mo stretchy="false">)</mo></mrow></msup></mrow></msubsup><mo>,</mo><msubsup><mi>J</mi> <mi>jk</mi> <mi>C</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>δ</mi> <mrow><mi>i</mi><mo stretchy="false">[</mo><mi>j</mi></mrow></msub><msubsup><mi>K</mi> <mrow><mi>k</mi><mo stretchy="false">]</mo></mrow> <mi>C</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>C</mi></msubsup><mo>,</mo><msubsup><mi>J</mi> <mi>kl</mi> <mi>C</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>i</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>δ</mi> <mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow></msub><msubsup><mi>J</mi> <mrow><mi>j</mi><mo stretchy="false">]</mo><mi>l</mi></mrow> <mi>C</mi></msubsup><mo>−</mo><msub><mi>δ</mi> <mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">|</mo><mi>l</mi><mo stretchy="false">|</mo></mrow></msub><msubsup><mi>J</mi> <mrow><mi>j</mi><mo stretchy="false">]</mo><mi>k</mi></mrow> <mi>C</mi></msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>K</mi> <mi>i</mi> <mi>C</mi></msubsup><mo>,</mo><msubsup><mi>P</mi> <mi>j</mi> <mi>C</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>δ</mi> <mi>ij</mi></msub><msubsup><mi>P</mi> <mn>0</mn> <mi>C</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>C</mi></msubsup><mo>,</mo><msubsup><mi>P</mi> <mi>k</mi> <mi>C</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>i</mi><msub><mi>δ</mi> <mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow></msub><msubsup><mi>P</mi> <mrow><mi>j</mi><mo stretchy="false">]</mo></mrow> <mi>C</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>C</mi></msubsup><mo>,</mo><msubsup><mi>Q</mi> <mi>α</mi> <mi>C</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>σ</mi> <mi>ij</mi></msub><msubsup><mo stretchy="false">)</mo> <mi>α</mi> <mi>β</mi></msubsup><msubsup><mi>Q</mi> <mi>β</mi> <mi>C</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>C</mi></msubsup><mo>,</mo><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover> <mi>C</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mover><mi>σ</mi><mo stretchy="false">¯</mo></mover> <mi>ij</mi></msub><msubsup><mo stretchy="false">)</mo> <mover><mi>α</mi><mo>˙</mo></mover> <mover><mi>β</mi><mo>˙</mo></mover></msubsup><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>β</mi><mo>˙</mo></mover> <mi>C</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><msubsup><mi>Q</mi> <mi>α</mi> <mi>C</mi></msubsup><mo>,</mo><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover> <mi>C</mi></msubsup><mo stretchy="false">}</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msup><mi>σ</mi> <mn>0</mn></msup><msub><mo stretchy="false">)</mo> <mrow><mi>α</mi><mover><mi>α</mi><mo>˙</mo></mover></mrow></msub><msubsup><mi>P</mi> <mn>0</mn> <mi>C</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{lcl} [K^{^{(\rm{C})}}_i ,J^{C}_{jk} ] &=& - i \delta_{i[j} K^{C}_{k]} \\ [J^{C}_{ij} ,J^{C}_{kl} ] &=& i \big( \delta_{[i|k|} J^{C}_{j]l} - \delta_{[i|l|} J^{C}_{j]k} \big) \\ [K^{C}_i ,P^{C}_j ] &=& -i \delta_{ij} P^{C}_0 \\ [J^{C}_{ij} ,P^{C}_k ] &=& i \delta_{[i|k|}P^{C}_{j]} \\ [J^{C}_{ij}, Q^{C}_{\alpha} ] &=& (\sigma_{ij} )_{\alpha} ^{\beta} Q^{C}_{\beta} \\ [J^{C}_{ij}, \bar{Q}^{C}_{\dot\alpha} ] &=& (\bar{\sigma }_{ij})_{\dot\alpha}^{\dot\beta} \bar{Q}^{C}_{\dot\beta} \\ \{Q^{C}_{\alpha},\bar{Q}^{C}_{\dot\alpha}\} &=& -(\sigma^0 )_{\alpha\dot\alpha} P^{C}_0 \,. \end{array} </annotation></semantics></math></div> <p>The <strong>super-Galilean</strong> algebra is defined with generators</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.8em" minsize="1.8em">{</mo><msubsup><mi>K</mi> <mi>i</mi> <mi>G</mi></msubsup><mo>=</mo><mfrac><mn>1</mn><mi>c</mi></mfrac><msub><mi>J</mi> <mrow><mn>0</mn><mi>i</mi></mrow></msub><mo>,</mo><msubsup><mi>P</mi> <mn>0</mn> <mi>G</mi></msubsup><mo>=</mo><mi>c</mi><msub><mi>P</mi> <mn>0</mn></msub><mo>,</mo><msubsup><mi>Q</mi> <mi>α</mi> <mi>G</mi></msubsup><mo>=</mo><msub><mi>Q</mi> <mi>α</mi></msub><mo>,</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>G</mi></msubsup><mo>=</mo><msub><mi>J</mi> <mi>ij</mi></msub><mo>,</mo><msubsup><mi>P</mi> <mi>i</mi> <mi>G</mi></msubsup><mo>=</mo><msub><mi>P</mi> <mi>i</mi></msub><mo>,</mo><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover> <mi>G</mi></msubsup><mo>=</mo><msub><mi>Q</mi> <mover><mi>α</mi><mo>˙</mo></mover></msub><mo maxsize="1.8em" minsize="1.8em">}</mo></mrow><annotation encoding="application/x-tex"> \Big\{ K^{G}_i = \frac{1}{c} J_{0i} , P^{G}_0 = c P_0 , Q^{G}_{\alpha} = Q_{\alpha}, J^{G}_{ij} = J_{ij}, P^{G}_{i} = P_{i}, \bar{Q}^{G}_{\dot\alpha} = Q_{\dot\alpha} \Big\} </annotation></semantics></math></div> <p>and the following non-zero commutators:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left center left"><mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>K</mi> <mi>i</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mi>J</mi> <mi>jk</mi> <mi>G</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msub><mi>δ</mi> <mrow><mi>i</mi><mo stretchy="false">[</mo><mi>j</mi></mrow></msub><msubsup><mi>K</mi> <mrow><mi>k</mi><mo stretchy="false">]</mo></mrow> <mi>G</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mi>J</mi> <mi>kl</mi> <mi>G</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>i</mi><mo stretchy="false">(</mo><msub><mi>δ</mi> <mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow></msub><msubsup><mi>J</mi> <mrow><mi>j</mi><mo stretchy="false">]</mo><mi>l</mi></mrow> <mi>G</mi></msubsup><mo>−</mo><msub><mi>δ</mi> <mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">|</mo><mi>l</mi><mo stretchy="false">|</mo></mrow></msub><msubsup><mi>J</mi> <mrow><mi>j</mi><mo stretchy="false">]</mo><mi>k</mi></mrow> <mi>G</mi></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>K</mi> <mi>i</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mi>P</mi> <mn>0</mn> <mi>G</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>i</mi><msubsup><mi>P</mi> <mi>i</mi> <mi>G</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mi>P</mi> <mi>k</mi> <mi>G</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>i</mi><msub><mi>δ</mi> <mrow><mo stretchy="false">[</mo><mi>i</mi><mo stretchy="false">|</mo><mi>k</mi><mo stretchy="false">|</mo></mrow></msub><msubsup><mi>P</mi> <mrow><mi>j</mi><mo stretchy="false">]</mo></mrow> <mi>G</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mi>Q</mi> <mi>α</mi> <mi>G</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>σ</mi> <mi>ij</mi></msub><msubsup><mo stretchy="false">)</mo> <mi>α</mi> <mi>β</mi></msubsup><msubsup><mi>Q</mi> <mi>β</mi> <mi>G</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msubsup><mi>J</mi> <mi>ij</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover> <mi>G</mi></msubsup><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mover><mi>σ</mi><mo stretchy="false">¯</mo></mover> <mi>ij</mi></msub><msubsup><mo stretchy="false">)</mo> <mover><mi>α</mi><mo>˙</mo></mover> <mover><mi>β</mi><mo>˙</mo></mover></msubsup><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>β</mi><mo>˙</mo></mover> <mi>G</mi></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><msubsup><mi>Q</mi> <mi>α</mi> <mi>G</mi></msubsup><mo>,</mo><msubsup><mover><mi>Q</mi><mo stretchy="false">¯</mo></mover> <mover><mi>α</mi><mo>˙</mo></mover> <mi>G</mi></msubsup><mo stretchy="false">}</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><msup><mi>σ</mi> <mi>i</mi></msup><msub><mo stretchy="false">)</mo> <mrow><mi>α</mi><mover><mi>α</mi><mo>˙</mo></mover></mrow></msub><msubsup><mi>P</mi> <mi>i</mi> <mi>G</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{lcl} [K^{G}_i ,J^{G}_{jk} ] &=& -i \delta_{i[j} K^{G}_{k]} \\ [J^{G}_{ij}, J^{G}_{kl} ] &=& i(\delta_{[i|k|} J^{G}_{j]l} - \delta_{[i|l|}J^{G}_{j]k} ) \\ [K^{G}_i ,P^{G}_0 ] &=& - i P^{G}_i \\ [J^{G}_{ij}, P^{G}_k ] &=& i \delta_{[i|k|} P^{G}_{j]} \\ [J^{G}_{ij},Q^{G}_{\alpha} ] &=& (\sigma_{ij} )_{\alpha}^{\beta} Q^{G}_{\beta} \\ [J^{G}_{ij}, \bar{Q}^{G}_{\dot\alpha} ] &=& (\bar{\sigma}_{ij})_{\dot\alpha}^{\dot\beta} \bar{Q}^{G}_{\dot\beta} \\ \{Q^{G}_{\alpha},\bar{Q}^{G}_{\dot\alpha}\} &=& -(\sigma^i)_{\alpha\dot\alpha} P^{G}_i \,. \end{array} </annotation></semantics></math></div> <p>See <a href="#BFG23">Bergshoeff et al. 2023</a> and the references therein for more.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a>, <a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a>, <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a>, <a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <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/R-symmetry">R-symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+translation+algebra">super translation algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a>, <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4d+supergravity+Lie+2-algebra">4d supergravity Lie 2-algebra</a>, <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="references">References</h2> <p>Introducing the <a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a> (“<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>”):</p> <ul> <li id="GolfandLikhtman72"> <p><a class="existingWikiWord" href="/nlab/show/Yuri+Golfand">Yuri Golfand</a>, <a class="existingWikiWord" href="/nlab/show/Evgeny+Likhtman">Evgeny Likhtman</a>,_On the Extensions of the Algebra of the Generators of the Poincaré Group by the Bispinor Generators_, in: <a class="existingWikiWord" href="/nlab/show/Victor+Ginzburg">Victor Ginzburg</a> et al. (eds.) <em>I. E. Tamm Memorial Volume Problems of Theoretical Physics</em>, (Nauka, Moscow 1972), page 37,</p> <p>translated and reprinted in: <a class="existingWikiWord" href="/nlab/show/Mikhail+Shifman">Mikhail Shifman</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/The+Many+Faces+of+the+Superworld">The Many Faces of the Superworld</a></em> pp. 44-53, World Scientific (2000) (<a href="https://doi.org/10.1142/9789812793850_0006">doi:10.1142/9789812793850_0006</a>)</p> </li> </ul> <h3 id="general">General</h3> <ul> <li id="CastellaniDAuriaFre"><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.2.1 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)</li> </ul> <p>The seminal classification result of simple supersymmetry algebras is due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Werner+Nahm">Werner Nahm</a>, <em>Supersymmetries and their Representations</em>, Nucl.Phys. B135 (1978) 149 (<a href="http://inspirehep.net/record/120988?ln=en">spire</a>)</li> </ul> <p>Lecture notes include</p> <ul> <li> <p><em>Super spacetimes and super Poincaré-group</em> (<a href="http://www.math.ucla.edu/~vsv/papers/ch6.pdf">pdf</a>)</p> </li> <li id="Freed01"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, lecture 6 of <em>Classical field theory and Supersymmetry</em>, IAS/Park City Mathematics Series Volume 11 (2001) (<a href="https://www.ma.utexas.edu/users/dafr/pcmi.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Lecture 4 of <a class="existingWikiWord" href="/nlab/show/Five+lectures+on+supersymmetry">Five lectures on supersymmetry</a></em></p> </li> <li id="Varadarajan04"> <p><a class="existingWikiWord" href="/nlab/show/Veeravalli+Varadarajan">Veeravalli Varadarajan</a>, section 7 of <em><a class="existingWikiWord" href="/nlab/show/Supersymmetry+for+mathematicians">Supersymmetry for mathematicians</a>: An introduction</em>, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)</p> </li> </ul> <p>See also</p> <ul> <li id="CAIB99">C. Chrysso‌malakos, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Izquierdo">José M. Izquierdo</a>, C. Pérez Bueno, <em>The geometry of branes and extended superspaces</em>, Nucl. Phys. B <strong>567</strong> (2000) 293-330 [<a href="http://arxiv.org/abs/hep-th/9904137">arXiv:hep-th/9904137</a>, <a href="https://doi.org/10.1016/S0550-3213(99)00512-X">doi:10.1016/S0550-3213(99)00512-X</a>]</li> </ul> <p>for discussion in the view of the <a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a> and <a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a> of super-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/brane">brane</a> <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma-models">Green-Schwarz sigma-models</a>.</p> <h3 id="PolyvectorExtensionRefs">Polyvector extensions</h3> <p>The Polyvector extensions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mn>10</mn><mo>,</mo><mn>1</mn><mo stretchy="false">|</mo><mn>32</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Iso}(\mathbb{R}^{10,1|32})</annotation></semantics></math> (the “<a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>”) were first considered in</p> <ul> <li id="DAuriaFre82"> <p><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a> <em><a class="existingWikiWord" href="/nlab/files/GeometricSupergravity.pdf" title="Geometric Supergravity in D=11 and its hidden supergroup">Geometric Supergravity in D=11 and its hidden supergroup</a></em>, Nuclear Physics B201 (1982) 101-140</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jan-Willem+van+Holten">Jan-Willem van Holten</a>, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em><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> supersymmetry algebras in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>8</mn></mrow><annotation encoding="application/x-tex">d=2,3,4 \,mod\, 8</annotation></semantics></math></em> J.Phys. A15, 3763 (1982).</p> </li> </ul> <p>Polyvector extensions were found as the algebra of <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+super+p-branes">Green-Schwarz super p-branes</a> in</p> <ul> <li id="AGIT89"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jerome+Gauntlett">Jerome Gauntlett</a>, J.M. Izquierdo, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Topological Extensions of the Supersymmetry Algebra for Extended Objects</em>, Phys.Rev.Lett. 63 (1989) 2443 (<a href="https://inspirehep.net/record/26393?ln=en">spire</a>)</li> </ul> <p>reviewed in section 8.8. of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, José M. Izquierdo, <em>Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics</em> , Cambridge monographs of mathematical physics, (1995)</li> </ul> <p>and specifically for super-<a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> this discussion is in</p> <ul> <li>Hanno Hammer, <em>Topological Extensions of Noether Charge Algebras carried by D-p-branes</em>, Nucl.Phys. B521 (1998) 503-546 (<a href="http://arxiv.org/abs/hep-th/9711009">arXiv:hep-th/9711009</a>)</li> </ul> <p>The role of polyvector extended supersymmetry algebras in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> and <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> is further highlighted in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Brane Democracy</em> (<a href="http://arxiv.org/abs/hep-th/9507048">arXiv:hep-th/9507048</a>)</li> </ul> <p>A comprehensive account and classification of the polyvector extensions of the super Poincaré Lie algebras is in</p> <ul> <li id="ACDP"><a class="existingWikiWord" href="/nlab/show/Dmiti+Alekseevsky">Dmiti Alekseevsky</a>, <a class="existingWikiWord" href="/nlab/show/Vicente+Cort%C3%A9s">Vicente Cortés</a>, C. Devchand, <a class="existingWikiWord" href="/nlab/show/Antoine+Van+Proeyen">Antoine Van Proeyen</a>, <em>Polyvector Super-Poincaré Algebras</em> Commun.Math.Phys. 253 (2004) 385-422 (<a href="http://arxiv.org/abs/hep-th/0311107">arXiv:hep-th/0311107</a>)</li> </ul> <h3 id="ReferencesLieAlgebraCohomology">Super Lie algebra cohomology</h3> <p>Discussion of the super-<a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a> goes back to work on <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma+models">Green-Schwarz sigma models</a> in</p> <ul> <li id="AzcarragaTownsend89"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Townsend">Paul Townsend</a>, <em>Superspace geometry and classification of supersymmetric extended objects</em>, Phys. Rev. Lett. <strong>62</strong> (1989) 2579-2582 [<a href="https://doi.org/10.1103/PhysRevLett.62.2579">doi:10.1103/PhysRevLett.62.2579</a>, <a href="https://inspirehep.net/literature/284635">spire:284635</a>]</li> </ul> <p>A rigorous classification of these cocycles was later given in</p> <ul> <li id="Brandt12-13"> <p><a class="existingWikiWord" href="/nlab/show/Friedemann+Brandt">Friedemann Brandt</a>, <em>Supersymmetry algebra cohomology</em></p> <p><em>I: Definition and general structure</em> J. Math. Phys.51:122302, 2010, (<a href="http://arxiv.org/abs/0911.2118">arXiv:0911.2118</a>)</p> <p><em>II: Primitive elements in 2 and 3 dimensions</em>, J. Math. Phys. 51 (2010) 112303 (<a href="http://arxiv.org/abs/1004.2978">arXiv:1004.2978</a>)</p> <p><em>III: Primitive elements in four and five dimensions</em>, J. Math. Phys. 52:052301, 2011 (<a href="http://arxiv.org/abs/1005.2102">arXiv:1005.2102</a>)</p> <p><em>IV: Primitive elements in all dimensions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">D=4</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>11</mn></mrow><annotation encoding="application/x-tex">D=11</annotation></semantics></math></em>, J. Math. Phys. 54, 052302 (2013) (<a href="http://arxiv.org/abs/1303.6211">arXiv:1303.6211</a>)</p> </li> </ul> <p>A classification of some special cases of signature/supersymmetry of this is also in the following (using a computer algebra system):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Movshev">Mikhail Movshev</a>, <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, Renjun Xu, <em>Homology of Lie algebra of supersymmetries</em> (<a href="http://arxiv.org/abs/1011.4731">arXiv:1011.4731</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Movshev">Mikhail Movshev</a>, <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, Renjun Xu, <em>Homology of Lie algebra of supersymmetries and of super Poincaré Lie algebra</em>, Nuclear Physics B Volume 854, Issue 2, 11 January 2012, Pages 483–503 (<a href="http://arxiv.org/abs/1106.0335">arXiv:1106.0335</a>)</p> </li> </ul> <p>For applications of this classification see also at <em><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functional">Green-Schwarz action functional</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/brane+scan">brane scan</a></em>.</p> <p>An introduction to the exceptional fermionic cocycles on the super Poincaré Lie algebra, and their description using <a class="existingWikiWord" href="/nlab/show/normed+division+algebras">normed division algebras</a>, are discussed here:</p> <ul id="BaezHuerta10"> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <em>Division algebras and supersymmetry I</em> (<a href="http://arxiv.org/abs/0909.0551">arXiv:0909.0551</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <em>Division algebras and supersymmetry II</em> (<a href="http://arxiv.org/abs/1003.3436">arXiv:1003.34360</a>)</p> </li> </ul> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Huerta">John Huerta</a>, <em>Division Algebras, Supersymmetry and Higher Gauge Theory</em>, (<a href="http://arxiv.org/abs/1106.3385">arXiv:1106.3385</a>)</li> </ul> <p>This subsumes some of the results in (<a href="#AzcarragaTownsend89">Azcárraga-Townend</a>)</p> <p>Discussion of the corresponding <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a> <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+extensions">L-∞ extensions</a> in the context of <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+action+functionals">Green-Schwarz action functionals</a> and <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a> is in</p> <ul> <li id="FSS13"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em></li> </ul> <p>A direct constructions of ordinary (Lie algebraic) extensions of the super Poincaré Lie algebra by means of <a class="existingWikiWord" href="/nlab/show/division+algebras">division algebras</a> is in</p> <ul> <li>Jerzy Lukierski, Francesco Toppan, <em>Generalized Space-time Supersymmetries, Division Algebras and Octonionic M-theory</em> (<a href="http://cbpfindex.cbpf.br/publication_pdfs/NF00102.2010_08_03_10_47_48.pdf">pdf</a>)</li> </ul> <p>For more on this see at <em><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></em>.</p> <p>On the notion of contraction used for non-Lorentzian limits:</p> <ul> <li id="IW53"><a class="existingWikiWord" href="/nlab/show/Erdal+%C4%B0n%C3%B6n%C3%BC">Erdal İnönü</a>, <a class="existingWikiWord" href="/nlab/show/Eugene+Wigner">Eugene Wigner</a>. <em>On the Contraction of Groups and Their Representations</em>. Proceedings of the National Academy of Sciences 39, no. 6 (1953): 510-524. (<a href="https://doi.org/10.1073/pnas.39.6.510">doi</a>).</li> </ul> <p>Discussion of the Carrollian- and Galilean limits:</p> <ul> <li id="KN23"> <p>Konstantinos Koutrolikos, Mojtaba Najafizadeh. <em>Super-Carrollian and super-Galilean field theories.</em> Physical Review D 108, no. 12 (2023): 125014. (<a href="https://doi.org/10.1103/PhysRevD.108.125014">doi</a>).</p> </li> <li id="BFG23"> <p><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, and <a class="existingWikiWord" href="/nlab/show/Joaquim+Gomis">Joaquim Gomis</a>. <em>A non-lorentzian primer.</em> SciPost Physics Lecture Notes (2023): 069. (<a href="https://doi.org/10.21468/SciPostPhysLectNotes.69">doi</a>).</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 7, 2024 at 15:33:33. See the <a href="/nlab/history/super+Poincar%C3%A9+Lie+algebra" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/super+Poincar%C3%A9+Lie+algebra" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/17825/#Item_3">Discuss</a><span class="backintime"><a href="/nlab/revision/super+Poincar%C3%A9+Lie+algebra/66" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/super+Poincar%C3%A9+Lie+algebra" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/super+Poincar%C3%A9+Lie+algebra" accesskey="S" class="navlink" id="history" rel="nofollow">History (66 revisions)</a> <a href="/nlab/show/super+Poincar%C3%A9+Lie+algebra/cite" style="color: black">Cite</a> <a href="/nlab/print/super+Poincar%C3%A9+Lie+algebra" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/super+Poincar%C3%A9+Lie+algebra" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>