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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="lie_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> <h4 id="symplectic_geometry">Symplectic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#_symplectic_manifold'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math>: symplectic manifold</a></li> <li><a href='#_poisson_manifold'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math>: Poisson manifold</a></li> <li><a href='#_courant_algebroid'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math>: Courant algebroid</a></li> </ul> <li><a href='#relation_to_other_concepts'>Relation to other concepts</a></li> <ul> <li><a href='#to_chernsimons_theory'>To <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons theory</a></li> <li><a href='#to_multisymplectic_geometry'>To multisymplectic geometry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/Lie+n-algebroid">Lie n-algebroid</a> is <em>symplectic</em> if it is equipped with a non-degenerate binary <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a>. This generalizes the notion of a <a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a> on a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, to which it reduces for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>symplectic Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-algebroid</strong> is a pair</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔞</mi><mo>,</mo><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathfrak{a}, \langle -,- \rangle) </annotation></semantics></math></div> <p>consisting of</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Lie+n-algebroid">Lie n-algebroid</a> <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">\mathfrak{a}</annotation></semantics></math>;</p> </li> <li> <p>a binary <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle- , - \rangle</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+2)</annotation></semantics></math></p> <p>(a closed element in the shifted elements of the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{a})</annotation></semantics></math>)</p> <p>which is non-degenerate.</p> </li> </ul> </div> <h2 id="properties">Properties</h2> <p>The <a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a> that witnesses this transgression is the Lagrangian of the corresponding <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> with <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">\mathfrak{a}</annotation></semantics></math> as its target space and the invariant polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,- \rangle</annotation></semantics></math> as the (<a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> of) its background <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>.</p> <h2 id="examples">Examples</h2> <h3 id="_symplectic_manifold"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n=0</annotation></semantics></math>: symplectic manifold</h3> <p>A 0-Lie algebroid is just a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <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>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> is the algebra of smooth functions on <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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CE(X) = C^\infty(X) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> of <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> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> W(X) = \Omega^\bullet(X) </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham algebra</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. A degree 2-<a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> on <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> is therefore a non-degenerate closed <a class="existingWikiWord" href="/nlab/show/differential+form">2-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^2(X)</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic 2-form</a>.</p> <div class="standout"> <p>A <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, being a pair</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X,\;\; \omega) </annotation></semantics></math></div> <p>consisting of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <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 a symplectic 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, is a symplectic Lie 0-algebroid.</p> </div> <h3 id="_poisson_manifold"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math>: Poisson manifold</h3> <p>For a <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> <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> with Poisson bivector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>∧</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi \in \Gamma(T X) \wedge \Gamma(T X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{a})</annotation></semantics></math> of the corresponding <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔞</mi><mo>:</mo><mo>=</mo><mi>𝔓</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{a} := \mathfrak{P}(X,\pi) </annotation></semantics></math></div> <p>is that of multi-vector fields on <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>, equipped with the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mi>π</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>Sch</mi></msub></mrow><annotation encoding="application/x-tex">d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}</annotation></semantics></math> given by the <a class="existingWikiWord" href="/nlab/show/Schouten+bracket">Schouten bracket</a>.</p> <p>If we work locally in coordinates then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{a})</annotation></semantics></math> is generated from degree 0 elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">x^i</annotation></semantics></math> and degree 1 elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\partial_i</annotation></semantics></math>. The differential is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mi>π</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>Sch</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{CE(\mathfrak{a})} = [\pi, -]_{Sch} \,. </annotation></semantics></math></div> <p>The Poisson tensor is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>:</mo><mo>=</mo><mi>π</mi><mo>=</mo><msup><mi>π</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mo>∂</mo> <mi>i</mi></msub><mo>∧</mo><msub><mo>∂</mo> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\nu := \pi = \pi^{i j} \partial_i \wedge \partial_j</annotation></semantics></math> and that this is a <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">Lie algebroid cocycle</a> is the fact that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow></msub><mi>π</mi><mo>=</mo><mo stretchy="false">[</mo><mi>π</mi><mo>,</mo><mi>π</mi><msub><mo stretchy="false">]</mo> <mi>Sch</mi></msub><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{CE(\mathfrak{a})} \pi = [\pi,\pi]_{Sch} = 0 \,. </annotation></semantics></math></div> <p>By definition the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{a})</annotation></semantics></math> is generated from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">x^i</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\partial_i</annotation></semantics></math> and their shifted partners <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}x^i</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mo>∂</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{d}\partial_i</annotation></semantics></math>. The differential here is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo stretchy="false">[</mo><mi>π</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>+</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_{W(\mathfrak{a})} = [\pi , - ] + \mathbf{d} \,. </annotation></semantics></math></div> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> that is in transgression with the cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>=</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">\nu = \pi</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mo>∂</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>inv</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i) \;\;\; \in inv(\mathfrak{a}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>One checks directly that the element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>cs</mi> <mi>ω</mi></msub><mo>=</mo><msup><mi>π</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mo>∂</mo> <mi>i</mi></msub><mo>∧</mo><msub><mo>∂</mo> <mi>j</mi></msub><mo>+</mo><msup><mi>x</mi> <mi>i</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mo>∂</mo> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> cs_\omega = \pi^{i j} \partial_i \wedge \partial_j + x^i \wedge \mathbf{d} \partial_i </annotation></semantics></math></div> <p>is a Chern-Simons transgression element for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>,</p> <p>i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow></msub><mi>cs</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">d_{W(\mathfrak{a})} cs(\omega) = \omega</annotation></semantics></math>. The restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cs</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">cs_\omega</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{a})</annotation></semantics></math> is evidently the Poisson tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>.</p> </div> <p>More details on this at <a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a>.</p> <div class="standout"> <p>For a <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> <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> with Poisson tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>=</mo><msup><mi>π</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msup><msub><mo>∂</mo> <mi>i</mi></msub><mo>∧</mo><msub><mo>∂</mo> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\pi = \pi^{i j} \partial_i \wedge \partial_j</annotation></semantics></math>, the pair</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔓</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ω</mi><mo>=</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msub><mo>∂</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\mathfrak{P}(X,\pi), \;\;\; \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i)) </annotation></semantics></math></div> <p>consisting of the <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔓</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{P}(X,\pi)</annotation></semantics></math> and of the <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> that is in transgression with its canonical 2-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi><mo>=</mo><mi>π</mi></mrow><annotation encoding="application/x-tex">\nu = \pi</annotation></semantics></math> (the Poisson tensor) is a symplectic <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>.</p> </div> <h3 id="_courant_algebroid"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math>: Courant algebroid</h3> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-symplectic manifold encodes and is encoded by the structure of a <a class="existingWikiWord" href="/nlab/show/Courant+algebroid">Courant algebroid</a>.</p> <p>A Courant 2algebroid over the point if given by a <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a> with the symplectic form being the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a>. The coresponding Poisson tensor is the canonical 3-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -, [-,-] \rangle</annotation></semantics></math> on a semisimple Lie algebra. The extension classified by this is the <a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a>.</p> <h2 id="relation_to_other_concepts">Relation to other concepts</h2> <h3 id="to_chernsimons_theory">To <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons theory</h3> <p>Since the symplectic form on a symplectic Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Algebroid may be understood <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theoretically</a> as an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> on an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid">L-∞ algebroid</a>, every symplectic Lie <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-algebroid serves as a <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> for an <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a>: this is <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a>.</p> <p>We have</p> <ul> <li>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a>;</li> <li>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Courant+sigma+model">Courant sigma model</a></li> <li>and so on.</li> </ul> <h3 id="to_multisymplectic_geometry">To multisymplectic geometry</h3> <p>There is also the closely related notion of <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a>. See</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Categorified Symplectic Geometry and the String Lie 2-Algebra</em>, (<a href="http://arxiv.org/abs/0901.4721">arXiv</a>)</li> </ul> <p>for some relations of this to the above situation for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math>. Essentially multisymplectic geometry studies the higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ary brackets induced from the binary graded symplectic form discussed here. The relation between these two pictures is the same as that between as studied in the context of <a class="existingWikiWord" href="/nlab/show/hemistrict+Lie+2-algebra">hemistrict Lie 2-algebra</a>s.</p> <p>An article with more details on this:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Courant algebroids from categorified symplectic geometry</em> (<a href="http://math.ucr.edu/~chris/2plectic-algebroid_DRAFT.pdf">pdf</a>).</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid">L-∞ algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> from binary and non-degenerate <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_1"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></th><th><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrated</a> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> = <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_2"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_3"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>d <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></th><th><a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-</a><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>/ <a class="existingWikiWord" href="/nlab/show/real+polarization">real polarization</a> <a class="existingWikiWord" href="/nlab/show/leaf">leaf</a></th><th>= <a class="existingWikiWord" href="/nlab/show/brane">brane</a></th><th><a class="existingWikiWord" href="/nlab/show/n-module">(n+1)-module</a> of <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> in <a class="existingWikiWord" href="/nlab/show/codimension">codimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_4"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math></th><th>discussed in:</th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a></td><td style="text-align: left;">–</td><td style="text-align: left;">ordinary <a class="existingWikiWord" href="/nlab/show/space+of+states+%28in+geometric+quantization%29">space of states (in geometric quantization)</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></em></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coisotropic+submanifold">coisotropic submanifold</a> (of underlying <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>)</td><td style="text-align: left;"><a href="Poisson+sigma-model#Branes">brane of Poisson sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a> = <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over <a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantiized</a> <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><em><a class="existingWikiWord" href="/nlab/show/extended+geometric+quantization+of+2d+Chern-Simons+theory">extended geometric quantization of 2d Chern-Simons theory</a></em></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+2-groupoid">symplectic 2-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/3-plectic+geometry">3-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dirac+structure">Dirac structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> in <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_5"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic n-groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">(n+1)-plectic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_0477fc87f3bab6c93955b29dd7c515f68e3108fd_6"><semantics><mrow><mi>d</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">d = n+1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(adapted from <a class="existingWikiWord" href="/nlab/show/Some+title+containing+the+words+%22homotopy%22+and+%22symplectic%22%2C+e.g.+this+one">Ševera 00</a>)</p></div> <h2 id="references">References</h2> <p>The notion originates somewhere in the school of <a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>‘s school of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categorial</a> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>. The first published appearance of the notion at least for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">0 \leq n \leq 3</annotation></semantics></math> is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitry+Roytenberg">Dmitry Roytenberg</a>, <em>Courant algebroids, derived brackets and even symplectic supermanifolds</em>, PhD thesis [<a href="http://arxiv.org/abs/math/9910078">arXiv:9910078</a>]</li> </ul> <p>A good writeup of this material is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dmitry+Roytenberg">Dmitry Roytenberg</a>, <em>On the structure of graded symplectic supermanifolds and Courant algebroids</em>, in: <a class="existingWikiWord" href="/nlab/show/Theodore+Voronov">Theodore Voronov</a> (ed.), <em>Quantization, Poisson Brackets and Beyond</em>, Contemp. Math. <strong>315</strong>, Amer. Math. Soc. (2002) [<a href="http://arxiv.org/abs/math/0203110">arXiv:0203110</a>, <a href="https://bookstore.ams.org/conm-315">ISBN:978-0-8218-7905-4</a>]</li> </ul> <p>The idea for all <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> was then sketched, together with many other ideas about <a class="existingWikiWord" href="/nlab/show/L-infinity+algebroid">L-infinity algebroid</a>s in the article with the nice title</p> <ul> <li><span class="newWikiWord">Pavol ?evera<a href="/nlab/new/Pavol+%3Fevera">?</a></span>, <em>Some title containing the words “homotopy” and “symplectic”, e.g. this one</em> (<a href="http://arxiv.org/abs/math/0105080">arXiv</a>)</li> </ul> <p>What we call <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>-symplectic manifold here is called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_n</annotation></semantics></math>-manifold there.</p> <p><strong>Warning</strong> This article here uses the term “<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>-symplectic” in a related but not identical sense to the one used here:</p> <ul> <li>M. de Leon, D. Martin de Diego, M. Salgado, S. Vilariño, <em>K-symplectic formalism on Lie algebroids</em> (<a href="http://lanl.arxiv.org/abs/0905.4585">arXiv</a>)</li> </ul> <p>A discussion of aspects of how <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> related to <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>-symplectic manifolds is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Courant algebroids from categorified symplectic geometry</em> (<a href="http://math.ucr.edu/~chris/2plectic-algebroid_DRAFT.pdf">pdf</a>)</li> </ul> <p><a href="http://front.math.ucdavis.edu/1001.0040">arXiv:1001.0040v1 [math-ph]</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Higher symplectic geometry</em> PhD thesis (2011) (<a href="http://arxiv.org/abs/1106.4068">arXiv:1106.4068</a>)</li> </ul> <p>A discussion of symplectic Lie n-algebroids from an <a class="existingWikiWord" href="/nlab/show/infinity-Lie+theory">infinity-Lie theory</a> perspective as discussed here is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/AKSZ+Sigma-Models+in+Higher+Chern-Weil+Theory">AKSZ Sigma-Models in Higher Chern-Weil Theory</a></em>, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (<a href="http://arxiv.org/abs/1108.4378">arXiv:1108.4378</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/H-cohomology">H-cohomology</a> of graded symplectic forms is considered in</p> <ul> <li id="Severa05"><a class="existingWikiWord" href="/nlab/show/Pavol+Severa">Pavol Severa</a>, p. 1 of <em>On the origin of the BV operator on odd symplectic supermanifolds</em>, Lett Math Phys (2006) 78: 55. (<a href="https://arxiv.org/abs/math/0506331">arXiv:0506331</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 17, 2022 at 15:43:46. 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