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href="/nlab/show/semigroup">semigroup</a>, <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Jordan+algebra">Jordan algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a>, <a class="existingWikiWord" href="/nlab/show/pre-Lie+algebra">pre-Lie algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> <li><a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></li> <li><a class="existingWikiWord" href="/nlab/show/commutator">commutator</a>, <a class="existingWikiWord" href="/nlab/show/center">center</a></li> <li><a class="existingWikiWord" href="/nlab/show/monad">monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></li> <li><a class="existingWikiWord" href="/nlab/show/distributive+law">distributive law</a></li> </ul> <h2 id="group_theory">Group theory</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/Cayley%27s+theorem">Cayley's theorem</a></li> <li><a class="existingWikiWord" href="/nlab/show/centralizer">centralizer</a>, <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, <a class="existingWikiWord" href="/nlab/show/Galois+extension">Galois extension</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></li> <li><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> <h2 id="ring_theory">Ring theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local ring</a>, <a class="existingWikiWord" href="/nlab/show/Artinian+ring">Artinian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noetherian+ring">Noetherian ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/skewfield">skewfield</a>, <a class="existingWikiWord" href="/nlab/show/field">field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+domain">integral domain</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal">ideal</a>, <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime ideal</a>, <a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ore+localization">Ore localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/central+simple+algebra">central simple algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>, <a class="existingWikiWord" href="/nlab/show/Ore+extension">Ore extension</a></p> </li> </ul> <h2 id="module_theory">Module theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/matrix">matrix</a>, <a class="existingWikiWord" href="/nlab/show/eigenvalue">eigenvalue</a>, <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/determinant">determinant</a>, <a class="existingWikiWord" href="/nlab/show/quasideterminant">quasideterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/Schur+lemma">Schur lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a>, <a class="existingWikiWord" href="/nlab/show/Morita+context">Morita context</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wedderburn-Artin+theorem">Wedderburn-Artin theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> </ul> <h2 id=""><a class="existingWikiWord" href="/nlab/show/gebra+theory">Gebras</a></h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a>, <a class="existingWikiWord" href="/nlab/show/coring">coring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comodule">comodule</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+module">Hopf module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yetter-Drinfeld+module">Yetter-Drinfeld module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associative+bialgebroid">associative bialgebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+gebra">dual gebra</a>, <a class="existingWikiWord" href="/nlab/show/cotensor+product">cotensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hopf-Galois+extension">Hopf-Galois extension</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> <h4 id="higher_spin_geometry">Higher spin geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/fivebrane+geometry">fivebrane geometry</a></strong> …</p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></p> </li> </ul> <h2 id="spin_geometry">Spin geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/central+product+spin+group">central product spin group</a></p> <p><a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor">spinor</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/charge+conjugation+matrix">charge conjugation matrix</a>, <a class="existingWikiWord" href="/nlab/show/Fierz+identity">Fierz identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/real+spin+representation">real spin representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+conjugate">Dirac conjugate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+spinor">Dirac spinor</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+spinor">Weyl spinor</a>, <a class="existingWikiWord" href="/nlab/show/Majorana+spinor">Majorana spinor</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+equation">Dirac equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/rotation+groups">rotation groups</a> in <a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low</a> <a class="existingWikiWord" href="/nlab/show/dimensions">dimensions</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/classification+of+simple+Lie+groups">Dynkin label</a></th><th><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">sp. orth. group</a></th><th><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></th><th><a class="existingWikiWord" href="/nlab/show/pin+group">pin group</a></th><th><a class="existingWikiWord" href="/nlab/show/semi-spin+group">semi-spin group</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%282%29">SO(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%282%29">Spin(2)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%282%29">Pin(2)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%283%29">SO(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%283%29">Spin(3)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%283%29">Pin(3)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%284%29">Spin(4)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%284%29">Pin(4)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%285%29">SO(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%285%29">Spin(5)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Pin%285%29">Pin(5)</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%286%29">SO(6)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%286%29">Spin(6)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B3</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%287%29">SO(7)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D4">D4</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%288%29">SO(8)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%288%29">Spin(8)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a href="semi-spin+group#SemiSpin8">SO(8)</a></td></tr> <tr><td style="text-align: left;">B4</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%289%29">SO(9)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%289%29">Spin(9)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D5">D5</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2810%29">SO(10)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2810%29">Spin(10)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">B5</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2811%29">SO(11)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2811%29">Spin(11)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D6">D6</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2812%29">SO(12)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2812%29">Spin(12)</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D8</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2816%29">SO(16)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2816%29">Spin(16)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2816%29">SemiSpin(16)</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</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><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">D16</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SO%2832%29">SO(32)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spin%2832%29">Spin(32)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/SemiSpin%2832%29">SemiSpin(32)</a></td></tr> </tbody></table> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Spin%285%29.Spin%283%29">Spin(5).Spin(3)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+rotation+groups">finite rotation groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ADE-classification">ADE-classification</a></p> </li> </ul> </div> <h2 id="string_geometry">String geometry</h2> <p><a class="existingWikiWord" href="/nlab/show/string+geometry">string geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+2-group">string^c 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/string%5Ec+structure">string^c structure</a></p> </li> </ul> <h2 id="fivebrane_geometry">Fivebrane geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></p> </li> </ul> <h2 id="ninebrane_geometry">Ninebrane geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#Classification'>Classification and Relation to matrix algebras</a></li> <ul> <li><a href='#ClassificationOverTheComplexNumbers'>Over the complex numbers</a></li> <li><a href='#ClassificationOverTheRealNumbers'>Over the real numbers</a></li> </ul> <li><a href='#as_a_superalgebra'>As a superalgebra</a></li> <li><a href='#RelationToOrthogonalLieAlgebras'>Relation to orthogonal Lie algebras</a></li> <li><a href='#relation_to__groups'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> groups</a></li> <li><a href='#AsQuantizedExteriorAlgebra'>Relation to exterior algebra (quantization)</a></li> </ul> <li><a href='#spinors'>Spinors</a></li> <li><a href='#warnings'>Warnings</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <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">R</annotation></semantics></math> and <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">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and <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>, an <strong><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">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/quadratic+function">quadratic function</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with values in <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> is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">q: M \to N</annotation></semantics></math> such that the following properties hold:</p> <ul> <li> <p>(cube relation) For any <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>M</mi></mrow><annotation encoding="application/x-tex">x,y,z \in M</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>q</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex">q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0.</annotation></semantics></math></div></li> <li> <p>(homogeneous of degree 2) For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x \in M</annotation></semantics></math> and any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">r \in R</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>r</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>r</mi> <mn>2</mn></msup><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">q(r x) = r^2q(x).</annotation></semantics></math></div></li> </ul> <p>A <strong>quadratic <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">R</annotation></semantics></math>-module</strong> is an <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">R</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> equipped with a <strong><a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a></strong>: an <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">R</annotation></semantics></math>-quadratic function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with values 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">R</annotation></semantics></math>.</p> <p>The <strong>Clifford algebra</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(M,q)</annotation></semantics></math> of a quadratic <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">R</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M,q)</annotation></semantics></math> can be defined as the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T_R(M)</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> generated by the relations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>x</mi><mo>−</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \otimes x - q(x)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x \in M</annotation></semantics></math>.</p> <p>Equivalently, it is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/object">object</a>s are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\phi)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an associative unital <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">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\phi: M \to A</annotation></semantics></math> is an <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">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mn>1</mn> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\phi(x)^2 = q(x) 1_A</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x \in M</annotation></semantics></math>, and whose morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>A</mi><mo>′</mo><mo>,</mo><mi>ϕ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\phi)\to (A',\phi')</annotation></semantics></math> are the associative <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">R</annotation></semantics></math>-algebra maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\chi: A\to A'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo>∘</mo><mi>ϕ</mi><mo>=</mo><mi>ϕ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\chi\circ\phi=\phi'</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <p>Examples in low <a class="existingWikiWord" href="/nlab/show/rank">rank</a> can be calculated easily. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is freely generated by a single element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>, with quadratic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">q(e) = 1</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mi>R</mi><mo stretchy="false">[</mo><mi>e</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>e</mi> <mn>2</mn></msup><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(M,q) = R[e]/(e^2-1)</annotation></semantics></math>. Note that the opposite sign convention is often used in the differential geometry literature, so one may see the assertion that the Clifford algebra of the real line with a positive definite metric is isomorphic to the complex numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mi>e</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>e</mi> <mn>2</mn></msup><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[e]/(e^2+1)</annotation></semantics></math>. Similarly, the Clifford algebra of a negative definite two dimensional real vector space is isomorphic to the (non-split) quaternions in our convention, but one may see the assertion that it is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M_2(\mathbb{R})</annotation></semantics></math>. Complexification removes the difference between positive definite and negative definite, and the two complexified algebras are isomorphic.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>=</mo><mi>L</mi><mo>⊕</mo><msup><mi>L</mi> <mo>∨</mo></msup></mrow><annotation encoding="application/x-tex">M = L \oplus L^\vee</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> projective of rank <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> over <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">R</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mo>∨</mo></msup><mo>=</mo><msub><mi>Hom</mi> <mi>R</mi></msub><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^\vee = Hom_R(L,R)</annotation></semantics></math> the dual module. One can define the canonical quadratic form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>f</mi><mo>+</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">q(f+x) = f(x)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>L</mi> <mo>∨</mo></msup></mrow><annotation encoding="application/x-tex">f \in L^\vee</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">x \in L</annotation></semantics></math>. In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>M</mi> <mrow><msup><mn>2</mn> <mi>d</mi></msup></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(M,q) \cong M_{2^d}(R)</annotation></semantics></math>. In general, the Clifford algebra arising from a nondegenerate form is flat-locally (on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Spec</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\operatorname{Spec} R</annotation></semantics></math>) isomorphic to a matrix algebra (when rank is even) or a direct sum of two matrix algebras (when rank is odd).</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a projective <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">R</annotation></semantics></math>-module of rank <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>, then independently of <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>, the Clifford algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(M,q)</annotation></semantics></math> is projective of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">2^d</annotation></semantics></math>, and is (noncanonically) isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="thinmathspace" rspace="thinmathspace">⋀</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\bigwedge M</annotation></semantics></math> as an <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">R</annotation></semantics></math>-module equipped with a map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. The Clifford algebra is isomorphic to the exterior algebra (as algebras equipped with <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">R</annotation></semantics></math>-module maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>) if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">q = 0</annotation></semantics></math>.</p> <p>If <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">R</annotation></semantics></math> is the ring of smooth functions on a pseudo-Riemannian manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is the <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">R</annotation></semantics></math>-module of sections of the tangent bundle, then the metric endows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with a quadratic structure, and one can form the Clifford algebra of the tangent bundle.</p> <h2 id="properties">Properties</h2> <h3 id="Classification">Classification and Relation to matrix algebras</h3> <h4 id="ClassificationOverTheComplexNumbers">Over the complex numbers</h4> <div class="num_theorem"> <h6 id="theorem">Theorem</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 the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> of complex <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>, equipped with non-degenerate <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a>, unique up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. The Clifford algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mi>d</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Cl_{d}(\mathbb{C}) \coloneqq Cl(V) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>, as a complex <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> to a <a class="existingWikiWord" href="/nlab/show/matrix+algebra">matrix algebra</a> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mi>d</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>Mat</mi> <mrow><msup><mn>2</mn> <mstyle displaystyle="false"><mfrac><mi>d</mi><mn>2</mn></mfrac></mstyle></msup></mrow></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>for</mi><mspace width="thinmathspace"></mspace><mi>d</mi><mspace width="thinmathspace"></mspace><mi>even</mi></mtd></mtr> <mtr><mtd><msub><mi>Mat</mi> <mrow><msup><mn>2</mn> <mstyle displaystyle="false"><mfrac><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mstyle></msup></mrow></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>⊕</mo><msub><mi>Mat</mi> <mrow><msup><mn>2</mn> <mstyle displaystyle="false"><mfrac><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mstyle></msup></mrow></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mtd> <mtd><mi>for</mi><mspace width="thinmathspace"></mspace><mi>d</mi><mspace width="thinmathspace"></mspace><mi>odd</mi></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> Cl_d(\mathbb{C}) \simeq \left\{ \array{ Mat_{2^{\tfrac{d}{2}}}(\mathbb{C}) & for \, d \, even \\ Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) \oplus Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) & for \, d \, odd } \right. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This is one of the incarnations of <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a>.</p> </div> <h4 id="ClassificationOverTheRealNumbers">Over the real numbers</h4> <p>We discuss the classification of Clifford algebras over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> and their relation to <a class="existingWikiWord" href="/nlab/show/matrix+algebras">matrix algebras</a> over the real numbers. A key statement is that of the mod-8 <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> of this classification (prop. <a class="maruku-ref" href="#RealBottPeriodicity"></a> below).</p> <p>In the following we write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Cl</mo> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Cl_{s,t}</annotation></semantics></math> for the Clifford algebra of a vector space over the real numbers with</p> <ul> <li> <p><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> generators squaring to -1</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> generators squaring to +1</p> </li> </ul> </li> <li> <p><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> for the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> regarded as an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> regarded as an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>≔</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><mi>𝕂</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{K}[n] \coloneqq Mat_{n \times n}(\mathbb{K})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/matrix+algebra">matrix algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">n \times n</annotation></semantics></math> matrices 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>𝕂</mi></mrow><annotation encoding="application/x-tex">\mathbb{K}</annotation></semantics></math>.</p> </li> </ul> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For low dimensions of real Clifford algebras, there are the following isomorphisms of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>≃</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> Cl_{0,1} \simeq \mathbb{R} \oplus \mathbb{R} </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>Cl</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>≃</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> Cl_{1,1} \simeq \mathbb{R}[2] </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>Cl</mi> <mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex"> Cl_{1,0} \simeq \mathbb{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>Cl</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex"> Cl_{2,0} \simeq \mathbb{H} </annotation></semantics></math></div></div> <p>(e. g. <a href="#FigueroaOFarrill">Figueroa-O’Farrill, lemma32</a>)</p> <div class="num_prop" id="Smooth0TypeIsSheavesOnSmoothMfd"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n,s,t \in \mathbb{N}</annotation></semantics></math> then there are the following <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msub><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msub><mi>Cl</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> Cl_{n+2,0} \simeq Cl_{0,n} \otimes_{\mathbb{R}} Cl_{2,0} </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>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>≃</mo><msub><mi>Cl</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msub><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> Cl_{0,n+2} \simeq Cl_{n,0} \otimes_{\mathbb{R}} Cl_{0,2} </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>Cl</mi> <mrow><mi>s</mi><mo>+</mo><mn>1</mn><mo>.</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≃</mo><msub><mi>Cl</mi> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msub><mi>Cl</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> Cl_{s+1.t+1} \simeq Cl_{s,t} \otimes_{\mathbb{R}} Cl_{1,1} </annotation></semantics></math></div></div> <p>(e.g. <a href="#LawsonMichelsohn89">Lawson-Michelsohn 89, theorem 4.1</a>, <a href="#FigueroaOFarrill">Figueroa-O’Farrill, lemma 2</a>)</p> <div class="num_prop" id="Smooth0TypeIsSheavesOnSmoothMfd"> <h6 id="proposition_3">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>,</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n,n_1, n_2 \in \mathbb{N}</annotation></semantics></math> there are the following <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>ℝ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>≃</mo><mi>ℝ</mi><mo stretchy="false">[</mo><msub><mi>n</mi> <mn>1</mn></msub><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[n_1] \otimes_{\mathbb{R}} \mathbb{R}[n_2] \simeq \mathbb{R}[n_1 n_2]</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>𝕂</mi><mo>≃</mo><mi>𝕂</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[n] \otimes_{\mathbb{R}} \mathbb{K} \simeq \mathbb{K}[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>ℂ</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>ℂ</mi><mo>≃</mo><mi>ℂ</mi><mo>⊕</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \simeq \mathbb{C} \oplus \mathbb{C}</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><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>ℍ</mi><mo>≃</mo><mi>ℂ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C} \otimes_{\mathbb{R}} \mathbb{H} \simeq \mathbb{C}[2]</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><mi>ℍ</mi><mo>≃</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mn>4</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H} \simeq \mathbb{R}[4]</annotation></semantics></math></p> </li> </ul> </div> <p>(e.g. <a href="#LawsonMichelsohn89">Lawson-Michelsohn 89, proposition 4.2</a>)</p> <p>Now the incarnation in Clifford algebras of <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> is the following:</p> <div class="num_prop" id="RealBottPeriodicity"> <h6 id="proposition_4">Proposition</h6> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> there are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a> over <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> as follows:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mrow><mi>n</mi><mo>+</mo><mn>8</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><msub><mi>Cl</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msub><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msub><mi>Cl</mi> <mrow><mn>8</mn><mo>,</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Cl_{n+8,0} \simeq Cl_{n,0} \otimes_{\mathbb{R}} Cl_{8,0}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mi>n</mi><mo>+</mo><mn>8</mn></mrow></msub><mo>≃</mo><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msub><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mn>8</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Cl_{0,n+8} \simeq Cl_{0,n} \otimes_{\mathbb{R}} Cl_{0,8}</annotation></semantics></math>.</p> </li> </ul> <p>where</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cl</mi> <mrow><mn>8</mn><mo>,</mo><mn>0</mn></mrow></msub><mo>≃</mo><msub><mi>Cl</mi> <mrow><mn>0</mn><mo>,</mo><mn>8</mn></mrow></msub><mo>≃</mo><mi>ℝ</mi><mo stretchy="false">[</mo><mn>16</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Cl_{8,0} \simeq Cl_{0,8} \simeq \mathbb{R}[16]</annotation></semantics></math>.</li> </ul> </div> <p>(e.g. <a href="#LawsonMichelsohn89">Lawson-Michelsohn 89, theorem 4.3</a>)</p> <h3 id="as_a_superalgebra">As a superalgebra</h3> <p>While the <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> of an <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">R</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> has a natural <a class="existingWikiWord" href="/nlab/show/integer">integer</a> grading, the quadratic relation collapses this to a natural <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 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(M,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><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><msup><mo stretchy="false">)</mo> <mi>ev</mi></msup><mo>⊕</mo><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><msup><mo stretchy="false">)</mo> <mi>odd</mi></msup></mrow><annotation encoding="application/x-tex"> Cl(M,q) = Cl(M,q)^{ev} \oplus Cl(M,q)^{odd} </annotation></semantics></math></div> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is projective of rank <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>, each homogeneous piece is projective of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>2</mn> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">2^{d-1}</annotation></semantics></math>. When <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> is nondegenerate, the even part of the Clifford algebra is also flat-locally isomorphic to a matrix ring or a sum of two matrix rings.</p> <p>One can view the Clifford algebra multiplication as a quantization of the commutative <a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋀</mo> <mi>R</mi></msub><mi>M</mi></mrow><annotation encoding="application/x-tex">\bigwedge_R M</annotation></semantics></math>.</p> <p><br /></p> <h3 id="RelationToOrthogonalLieAlgebras">Relation to orthogonal Lie algebras</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>Γ</mi> <mi>a</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{\Gamma_{a}\}_{a = 1}^n</annotation></semantics></math> be the generators of an <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>-dimensional <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> corresponding to an <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a>/metric <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">g</annotation></semantics></math>, hence with this <a class="existingWikiWord" href="/nlab/show/anti-commutator">anti-commutator</a>:</p> <div class="maruku-equation" id="eq:CliffordGeneratorsAnticommutator"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">{</mo><msub><mi>Γ</mi> <mi>a</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>b</mi></msub><mo maxsize="1.2em" minsize="1.2em">}</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>2</mn><msub><mi>g</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big\{ \Gamma_{a} ,\; \Gamma_b \big\} \;=\; 2 g_{a b} \,. </annotation></semantics></math></div> <p>Then:</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> via <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a>)</strong></p> <p>The elements of the Clifford algera <a class="maruku-eqref" href="#eq:CliffordGeneratorsAnticommutator">(1)</a> given by</p> <div class="maruku-equation" id="eq:CliffordLieAlgebra"><span class="maruku-eq-number">(2)</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><mi>r</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mo>≔</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><msub><mi>a</mi> <mn>1</mn></msub><mo>≠</mo><msub><mi>a</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} r_{a_1 a_2} & \coloneqq\; \tfrac{1}{4} \big[ \Gamma_{a_1}, \Gamma_{a_2} \big] \\ & = \left\{ \array{ \tfrac{1}{2} \Gamma_{a_1} \Gamma_{a_2} &\vert& a_1 \neq a_2 \\ 0 &\vert& otherwise } \right. \end{aligned} </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">a_1, a_2 \;\in\; \{1, \cdots, n\}</annotation></semantics></math>,</p> <p>and equipped with the <a class="existingWikiWord" href="/nlab/show/commutator+bracket">commutator bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>≔</mo><mi>a</mi><mi>b</mi><mo>−</mo><mi>b</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">[a,b] \coloneqq a b - b a</annotation></semantics></math></p> <p><a class="existingWikiWord" href="/nlab/show/linear+span">span</a> a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> which is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal 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>n</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(n,g)</annotation></semantics></math> with respect to <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">g</annotation></semantics></math>, in that their <a class="existingWikiWord" href="/nlab/show/commutator">commutator</a> <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> is:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>r</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo><msub><mi>r</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><msub><mi>r</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo>−</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><msub><mi>r</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo>+</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><msub><mi>r</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>−</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><msub><mi>r</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \big[ r_{a_1 a_2} \,, r_{b_1 b_2} \big] \;=\; g_{a_2 b_1} r_{a_1 b_2} - g_{a_1 b_1} r_{a_2 b_2} + g_{a_2 b_2} r_{b_1 a_1} - g_{a_1 b_2} r_{b_1 a_2} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>First observe that</p> <div class="maruku-equation" id="eq:CliffordLieActionOnVectorsInComponents"><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><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>b</mi></msub><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">{</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>b</mi></msub><mo maxsize="1.2em" minsize="1.2em">}</mo><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">{</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>b</mi></msub><mo maxsize="1.2em" minsize="1.2em">}</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub><mi>b</mi></mrow></msub><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>−</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>b</mi></mrow></msub><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \big[ \tfrac{1}{2} \Gamma_{a_1} \Gamma_{a_2} ,\; \Gamma_b \big] & = \tfrac{1}{2} \Gamma_{a_1} \big\{ \Gamma_{a_2}, \; \Gamma_b \big\} - \tfrac{1}{2} \big\{ \Gamma_{a_1}, \; \Gamma_b \big\} \Gamma_{a_2} \\ & = g_{a_2 b} \Gamma_{a_1} - g_{a_1 b} \Gamma_{a_2} \end{aligned} </annotation></semantics></math></div> <p>Here the first step may be thought of as the <em>graded</em> <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> in the Clifford <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, but it is also immediately verified by inspection. The second step evaluates the defining anti-commutators <a class="maruku-eqref" href="#eq:CliffordLieActionOnVectorsInComponents">(3)</a>.</p> <p>With this we compute as follows, assuming, without restriction of generality, that <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>2</mn></msub></mrow><annotation encoding="application/x-tex">a_1 \neq a_2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>b</mi> <mn>1</mn></msub><mo>≠</mo><msub><mi>b</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">b_1 \neq b_2</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 maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo>,</mo><mspace width="thickmathspace"></mspace><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd> <mtd><mo>=</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo maxsize="1.8em" minsize="1.8em">[</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo maxsize="1.8em" minsize="1.8em">]</mo><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo maxsize="1.8em" minsize="1.8em">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.8em" minsize="1.8em">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo>−</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo>+</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">]</mo><mo>−</mo><msub><mi>g</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>b</mi> <mn>2</mn></msub></mrow></msub><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \big[ \tfrac{1}{4} [\Gamma_{a_1}, \Gamma_{a_2}] , \; \tfrac{1}{4} [\Gamma_{b_1}, \Gamma_{b_2}] \big] & = \tfrac{1}{4} \Big[ \big[ \tfrac{1}{4} [\Gamma_{a_1},\Gamma_{a_2}] ,\, \Gamma_{b_1} \big] ,\, \Gamma_{b_2} \Big] + \tfrac{1}{4} \Big[ \Gamma_{b_1}, \big[ \tfrac{1}{4} [\Gamma_{a_1},\Gamma_{a_2}] ,\, \Gamma_{b_2} \big] \Big] \\ & = g_{a_2 b_1} \tfrac{1}{4} [\Gamma_{a_1},\Gamma_{b_2}] - g_{a_1 b_1} \tfrac{1}{4} [\Gamma_{a_2},\Gamma_{b_2}] + g_{a_2 b_2} \tfrac{1}{4} [\Gamma_{b_1},\Gamma_{a_1}] - g_{a_1 b_2} \tfrac{1}{4} [\Gamma_{b_1}, \Gamma_{a_2}] \end{aligned} </annotation></semantics></math></div> <p>Here the first step is again the graded <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> (and is again also immediely checked by inspection), while the second step uses <a class="maruku-eqref" href="#eq:CliffordLieActionOnVectorsInComponents">(3)</a>.</p> </div> <p><br /></p> <h3 id="relation_to__groups">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math> groups</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be a projective <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">R</annotation></semantics></math>-module of finite <a class="existingWikiWord" href="/nlab/show/rank">rank</a>, and let <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> be nondegenerate. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><msup><mo stretchy="false">)</mo> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">Cl(M,q)^\times</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a> of the Clifford algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(M,q)</annotation></semantics></math>.</p> <p>The <strong>Clifford group</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>M</mi><mo>,</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_{M,q}(R)</annotation></semantics></math> is the subgroup of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> for which twisted conjugation stabilizes the submodule <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M \subset Cl(M,q)</annotation></semantics></math>. Here, twisted conjugation is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>↦</mo><mi>x</mi><mi>y</mi><mi>α</mi><mo stretchy="false">(</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">y \mapsto x y\alpha(x)^{-1}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is the automorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CL</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CL(M,q)</annotation></semantics></math> induced by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math> map on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. Since twisted conjugation by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>-stabilizing elements amounts to reflection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>↦</mo><mi>y</mi><mo>−</mo><mn>2</mn><mfrac><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac><mi>x</mi></mrow><annotation encoding="application/x-tex">y \mapsto y - 2\frac{(x,y)}{q(x)}x</annotation></semantics></math>, there is a canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>M</mi><mo>,</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_{M,q}(R) \to O(M,q)</annotation></semantics></math>, and the Clifford group is in fact a central extension of the orthogonal group by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">R^\times</annotation></semantics></math>.</p> <p>The Clifford group is made of homogeneous elements in 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, and the subgroup of even elements is a normal subgroup of index two. One also has a <a class="existingWikiWord" href="/nlab/show/spinor">spinor</a> norm <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>:</mo><msub><mi>Γ</mi> <mrow><mi>M</mi><mo>,</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>R</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">Q: \Gamma_{M,q}(R) \to R^\times</annotation></semantics></math> on the Clifford group, defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi> <mi>t</mi></msup><mi>x</mi></mrow><annotation encoding="application/x-tex">Q(x) = x^t x</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><msup><mi>x</mi> <mi>t</mi></msup></mrow><annotation encoding="application/x-tex">x \mapsto x^t</annotation></semantics></math> is the anti-involution of the Clifford algebra defined by opposite multiplication in the tensor algebra.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Pin+group">Pin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Pin</mi> <mrow><mi>M</mi><mo>,</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pin_{M,q}(R)</annotation></semantics></math> is the group elements of the Clifford group with <a class="existingWikiWord" href="/nlab/show/spinor">spinor</a> norm 1. The <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Spin</mi> <mrow><mi>M</mi><mo>,</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin_{M,q}(R)</annotation></semantics></math> is the group of elements in the even subgroup of the Clifford group with <a class="existingWikiWord" href="/nlab/show/spinor">spinor</a> norm 1.</p> <p>The restriction of the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>M</mi><mo>,</mo><mi>q</mi></mrow></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>→</mo><mi>O</mi><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_{M,q}(R) \to O(M,q)</annotation></semantics></math> to the Pin group may not be surjective, but it is for positive definite real vector spaces. The kernel is the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_2(R)</annotation></semantics></math> of elements of <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">R</annotation></semantics></math> that square to 1. Similarly, the <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> has a map to the special orthogonal group with kernel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_2(R)</annotation></semantics></math>, but it may not be surjective in general.</p> <p>One can use base change to define the groups given above as functors on commutative <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">R</annotation></semantics></math>-algebras.</p> <h3 id="AsQuantizedExteriorAlgebra">Relation to exterior algebra (quantization)</h3> <p>For <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> an <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner product space</a>, the <a class="existingWikiWord" href="/nlab/show/symbol+map">symbol map</a> (see there) constitutes an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of the underlying <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>s of the Clifford algebra with the <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior 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>.</p> <p>One may understand the Clifford algebra as the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> induced from the <a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a> regarded as an odd <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>.</p> <h2 id="spinors">Spinors</h2> <p>For nondegenerate <a class="existingWikiWord" href="/nlab/show/quadratic+forms">quadratic forms</a> on real vector spaces, <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>/<a class="existingWikiWord" href="/nlab/show/spin+representations">spin representations</a> are distinguished linear representations of <a class="existingWikiWord" href="/nlab/show/Spin+groups">Spin groups</a> that are not pulled back from the corresponding special orthogonal groups. In other words, the central element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math> acts nontrivially. They can be realized as restrictions of representations of the even parts of Clifford algebras. Since even parts of Clifford algebras are (up to complexification) the sum of one or two matrix rings, their representation theory is quite simple.</p> <p>The specific nature of spinor representations possible depends on the signature of the vector space modulo 8. This is a manifestation of Bott periodicity. One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra. In even dimensions, this splits into two Weyl spinors (called half-spin representations). One may also have real representations called Majorana spinors, and these may decompose into Majorana-Weyl spinors.</p> <p>There are infinite dimensional Clifford algebra constructions that appear in conformal field theory. One may extend the above discussion to topological <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">R</annotation></semantics></math>-modules and continuous quadratic forms, and one obtains canonical central extensions of infinite dimensional groups and algebras by a relative determinant construction. Semi-infinite wedge spaces are spinor modules for Clifford algebras of quadratic Tate <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">R</annotation></semantics></math>-modules.</p> <h2 id="warnings">Warnings</h2> <p>There is a difference of sign convention between differential geometers (following Atiyah) and everyone else.</p> <p>Clifford algebras are often defined using bilinear forms instead of quadratic forms (and one often sees incorrect definitions of quadratic forms in terms of bilinear forms). Such definitions will yield wrong (or boring) objects when 2 is not invertible.</p> <p>Special orthogonal groups are often defined as the kernel of the determinant map on the corresponding orthogonal groups, but in characteristic 2, the determinant is trivial, while the Clifford grading (called the Dickson invariant) is not.</p> <p>The Clifford group is sometimes defined without the twist in the conjugation, and this means the map to the orthogonal group may not be a surjection, and the action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is by negative reflections.</p> <p>The spinor norm is sometimes defined with the opposite sign.</p> <p>Special orthogonal groups over the reals are sometimes defined to be the connected component of the identity in the orthogonal group. In indefinite signature, this defines an index two subgroup of the special orthogonal group.</p> <p>Spin groups in signature <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(m,n)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">m,n \geq 2</annotation></semantics></math> have fundamental groups of order two. They are simply connected when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> or <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> is at most one.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+algebra">geometric algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+module">Clifford module</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+bundle">Clifford bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+module+bundle">Clifford module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fierz+identity">Fierz identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a>, <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-Clifford+algebra">2-Clifford algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+slash+notation">Feynman slash notation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pythagorean+ring">Pythagorean ring</a></p> </li> </ul> <h2 id="references">References</h2> <p>The notion of Clifford algebra is due to</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Clifford">William Clifford</a>, <em>Applications of Grassmann’s extensive algebra</em>, American Journal of Mathematics <strong>1</strong> 4 (1878) 350-358 [<a href="https://doi.org/10.2307/2369379">doi:10.2307/2369379</a>, <a href="https://www.jstor.org/stable/2369379">jstor:2369379</a>]</p> <blockquote> <p>(referring to <a class="existingWikiWord" href="/nlab/show/Hermann+Grassmann">Hermann Grassmann</a>‘s <em><a class="existingWikiWord" href="/nlab/show/Ausdehnungslehre">Ausdehnungslehre</a></em>)</p> </blockquote> </li> </ul> <p>Further early discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Emil+Artin">Emil Artin</a>, §V.4 in: <em><a class="existingWikiWord" href="/nlab/show/Geometric+Algebra">Geometric Algebra</a></em>, Wiley 1957 (1988) [<a href="https://www.wiley.com/en-ae/Geometric+Algebra-p-9781118164549">ISBN:978-1-118-16454-9</a>, <a href="https://en.wikipedia.org/wiki/Geometric_Algebra_(book)">Wikipedia entry</a>, <a href="https://archive.org/details/geometricalgebra033556mbp/page/n5/mode/2upa">ark:/13960/t4nk37034</a>]</p> </li> <li id="Cartan66"> <p><a class="existingWikiWord" href="/nlab/show/%C3%89lie+Cartan">Élie Cartan</a>, <em>Theory of Spinors</em>, Dover, (1966)</p> </li> </ul> <p>Brief introductions:</p> <ul> <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>, Ch II.7 in: <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.7: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChII7.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+Gallier">Jean Gallier</a>, <em>Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions</em> (<a href="https://arxiv.org/abs/0805.0311">arXiv:0805.0311</a>)</p> </li> <li> <p>Marc Lachieze-Rey, <em>Spin and Clifford algebras, an introduction</em> (<a href="https://arxiv.org/abs/1007.2481">arXiv:1007.2481</a>)</p> </li> </ul> <p>Standard textbook accounts:</p> <ul> <li id="LawsonMichelsohn89"> <p><a class="existingWikiWord" href="/nlab/show/H.+Blaine+Lawson">H. Blaine Lawson</a>, <a class="existingWikiWord" href="/nlab/show/Marie-Louise+Michelsohn">Marie-Louise Michelsohn</a>, <em><a class="existingWikiWord" href="/nlab/show/Spin+geometry">Spin geometry</a></em>, Princeton University Press (1989) [<a href="https://press.princeton.edu/books/hardcover/9780691085425/spin-geometry-pms-38-volume-38">ISBN 9780691085425</a>]</p> </li> <li id="Lounesto"> <p><a class="existingWikiWord" href="/nlab/show/Pertti+Lounesto">Pertti Lounesto</a>, <em>Clifford Algebras and Spinors</em>, London Mathematical Society <strong>286</strong>, Cambridge University Press (2001) [<a href="https://doi.org/10.1017/CBO9780511526022">doi:10.1017/CBO9780511526022</a>]</p> </li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eckhard+Meinrenken">Eckhard Meinrenken</a>: <em>Clifford algebras and Lie groups</em>, Ergebn. Mathem. & Grenzgeb., Springer (2013) [<a href="https://doi.org/10.1007/978-3-642-36216-3">doi:10.1007/978-3-642-36216-3</a>]</li> </ul> <p>For a program that promotes the use of Clifford algebra as a good expositional tool in introductory <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> see <em><a class="existingWikiWord" href="/nlab/show/Geometric+Algebra">Geometric Algebra</a></em>.</p> <p>See also the discussion of <a class="existingWikiWord" href="/nlab/show/Majorana+spinors">Majorana spinors</a></p> <ul> <li id="FigueroaOFarrill"><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <em>Majorana spinors</em> (<a href="http://www.maths.ed.ac.uk/~jmf/Teaching/Lectures/Majorana.pdf">pdf</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/subgroups">subgroups</a> of Clifford algebras reminiscent of <a class="existingWikiWord" href="/nlab/show/gauge+groups">gauge groups</a> of relevance in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> and <a class="existingWikiWord" href="/nlab/show/grand+unified+theories">grand unified theories</a> thereof:</p> <ul> <li id="Wilson20"> <p><a class="existingWikiWord" href="/nlab/show/Robert+A.+Wilson">Robert A. Wilson</a>, <em>A group-theorist’s perspective on symmetry groups in physics</em> (<a href="https://arxiv.org/abs/2009.14613">arXiv:2009.14613</a>)</p> </li> <li id="Wilson21"> <p><a class="existingWikiWord" href="/nlab/show/Robert+A.+Wilson">Robert A. Wilson</a>, <em>On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics</em>, Adv. Appl. Clifford Algebras 31, 59 (2021) (<a href="https://doi.org/10.1007/s00006-021-01160-5">doi:10.1007/s00006-021-01160-5</a>)</p> </li> </ul> <p>(For similar investigations, see also <a href="Albert+algebra#ReferencesRelationStandardModel">here</a> at <em><a class="existingWikiWord" href="/nlab/show/Albert+algebra">Albert algebra</a></em>.)</p> </body></html> </div> <div class="revisedby"> <p> Last revised on September 11, 2024 at 09:43:17. 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