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geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="rational_homotopy_theory">Rational homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+object">differential</a> <a class="existingWikiWord" href="/nlab/show/graded+object">graded objects</a></strong></p> <p>and</p> <p><strong><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/equivariant+rational+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/parametrized+rational+homotopy+theory">parametrized</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+rational+stable+homotopy+theory">equivariant & stable</a>, <a class="existingWikiWord" href="/nlab/show/parametrized+rational+stable+homotopy+theory">parametrized & stable</a>)</p> <h2 id="dgalgebra">dg-Algebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+vector+space">differential graded vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">differential graded algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+coalgebra">differential graded coalgebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+Lie+algebra">differential graded Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+Hopf+algebra">differential graded Hopf algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bar+and+cobar+construction">bar and cobar construction</a></p> </li> </ul> <h2 id="rational_spaces">Rational spaces</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nilpotent+space">nilpotent space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+space">rational space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+space">formal space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rationalization">rationalization</a></p> </li> </ul> <h2 id="pl_de_rham_complex">PL de Rham complex</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+on+simplices">differential forms on simplices</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PL+de+Rham+complex">PL de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+construction">Sullivan construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+dg-algebraic+rational+homotopy+theory">fundamental theorem of dg-algebraic rational homotopy theory</a></p> </li> </ul> <h2 id="sullivan_models">Sullivan models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/minimal+Sullivan+model">minimal</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Sullivan+model">Sullivan model</a></li> </ul> <div> <p><strong>Examples of <a class="existingWikiWord" href="/nlab/show/Sullivan+models">Sullivan models</a></strong> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+an+n-sphere">Sullivan model of an n-sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+spherical+fibration">Sullivan model of a spherical fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+complex+projective+space">Sullivan model of complex projective space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+classifying+space">Sullivan model of a classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+mapping+space">Sullivan model of mapping space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+free+loop+space">Sullivan model of free loop space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+suspension">Sullivan model of a suspension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sullivan+model+of+a+finite+G-quotient">Sullivan model of a finite G-quotient</a></p> </li> </ul> </div> <h2 id="related_topics">Related topics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#History'>History</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_terms_of_algebras_over_an_operad'>In terms of algebras over an operad</a></li> <li><a href='#DefinitionViaHigherBrackets'>In terms of higher brackets</a></li> <li><a href='#ReformulationInSemifreeDgCoalgebra'>In terms of semifree differential coalgebra</a></li> <li><a href='#ReformulationInTermsOfSemifreeDGAlgebra'>In terms of semifree differential algebra</a></li> <ul> <li><a href='#DGAlgebraDetails'>Details</a></li> </ul> <li><a href='#InTermsOfAlgebrasOverAnOperad'>In terms of algebras over an operad</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#special_cases'>Special cases</a></li> <li><a href='#classes_of_examples'>Classes of examples</a></li> <li><a href='#specific_examples'>Specific examples</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#IndConilpotency'>Ind-Conilpotency</a></li> <li><a href='#model_category_structure'>Model category structure</a></li> <li><a href='#relation_to_dglie_algebras'>Relation to dg-Lie algebras</a></li> <li><a href='#relation_to_lie_groupoids'>Relation 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>-Lie groupoids</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#as_models_for_rational_homotopy_types'>As models for rational homotopy types</a></li> <li><a href='#ReferencesInPhysics'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras in physics</a></li> <ul> <li><a href='#in_supergravity'>In supergravity</a></li> <li><a href='#ReferencesCFieldGaugeAlgebra'>Supergravity C-Field gauge algebra</a></li> <li><a href='#ReferencesBVBRSTFormalism'>In BV-BRST formalism</a></li> <li><a href='#in_string_field_theory'>In string field theory</a></li> <li><a href='#in_deformation_quantization'>In deformation quantization</a></li> <li><a href='#in_heterotic_string_theory'>In heterotic string theory</a></li> <li><a href='#higher_chernsimons_field_theory_and_aksz_sigmamodels'>Higher Chern-Simons field theory and AKSZ sigma-models</a></li> <li><a href='#in_local_prequantum_field_theory'>In local prequantum field theory</a></li> <li><a href='#in_perturbative_quantum_field_theory'>In perturbative quantum field theory</a></li> <li><a href='#in_double_field_theory'>In double field theory</a></li> </ul> <li><a href='#related_expositions'>Related expositions</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras</em> (or <em>strong homotopy Lie algebras</em>) are a higher generalization (a “<a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a>”) of <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>: in an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> is allowed to hold (only) up to higher <a class="existingWikiWord" href="/nlab/show/coherence+law">coherent</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>.</p> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra that is concentrated in lowest degree is an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. If it is concentrated in the lowest two degrees is is a <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a>, etc.</p> <p>From another perspective: an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra is a <a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid">Lie ∞-algebroid</a> with a single object.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras are <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> approximations of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a>s in analogy to how an ordinary Lie algebra is an infinitesimal approximation of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. Under <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> “exponentiates” to a <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>exp</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega \exp(\mathfrak{g})</annotation></semantics></math>.</p> <h2 id="History">History</h2> <div class="num_remark" id="SuperLInfintiyAsFDA"> <h6 id="remark">Remark</h6> <p><strong>(history of the concept of (super-)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math> algebras)</strong></p> <p>The identification of the concept of (super-)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras has a non-linear history:</p> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> in the incarnation of higher brackets satisfying a higher Jacobi identity (def. <a class="maruku-ref" href="#LInfinityDefinitionViaGeneralizedJacobiIdentity"></a>) were introduced in <a href="#Stasheff92">Stasheff 92</a>, <a href="#LadaStasheff92">Lada-Stasheff 92</a>, based on the example of such a structure on the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> that was reported in the construction of <a class="existingWikiWord" href="/nlab/show/closed+string+field+theory">closed string field theory</a> in <a href="#Zwiebach92">Zwiebach 92</a>.</p> <p><a href="#LadaStasheff92">Lada-Stasheff 92</a> credit <a href="deformation+theory#SchlessingerStasheff85">Schlessinger-Stasheff 85</a> with the introduction of the concept, but while that article considers many closely related structures, it does not consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras as such. <a href="#LadaMarkl94">Lada-Markl 94</a> credit other work by Schlessinger-Stasheff as the origin, but that work appeared much later as <a href="deformation+theory#SchlessingerStasheff12">Schlessinger-Stasheff 12</a>.</p> <p>According to <a href="#Stasheff16">Stasheff 16, slide 25</a>, <a class="existingWikiWord" href="/nlab/show/Barton+Zwiebach">Zwiebach</a> had this structure already in 1989, when <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Stasheff</a> recognized it in a talk by Zwiebach at a <a class="existingWikiWord" href="/nlab/show/GUT">GUT</a> conference in Chapel-Hill. Zwiebach, in turn, was following the <a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> of <a href="#BatalinVilkovisky83">Batalin-Vilkovisky 83</a>, <a href="#BatakinFradkin83">Batakin-Fradkin 83</a>, whose relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras was later amplified in <a href="#Stasheff96">Stasheff 96</a>, <a href="#Stasheff97">Stasheff 97</a>.</p> <p>The observation that these systems of higher brackets are fully characterized by their Chevalley-Eilenberg dg-(co-)algebras is due to <a href="#LadaMarkl94">Lada-Markl 94</a>. See <a href="#SatiSchreiberStasheff08">Sati-Schreiber-Stasheff 08, around def. 13</a>.</p> <p>But in this dual incarnation, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> and more generally <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a> (of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a>) had secretly been introduced, independently of the <a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> of <a href="#BatalinVilkovisky83">Batalin-Vilkovisky 83</a>, <a href="#BatakinFradkin83">Batakin-Fradkin 83</a>, within the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature already in <a href="D'Auria-Fré-Regge+formulation+of+supergravity#DAuriaFréRegge80b">D’Auria-Fré-Regge 80</a> and explicitly in <a href="#Nieuwenhuizen82">van Nieuwenhuizen 82</a>. The concept was picked up in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> (<a href="#DAuriaFre82">D’Auria-Fré 82</a>) and eventually came to be referred to as “FDA”s (short for “free differential algebra”) in the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature (but beware that these dg-algebras are in general <a class="existingWikiWord" href="/nlab/show/free+construction">free</a> only as graded-<a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>, not as differential algebras) The relation between super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras and the “FDA”s of the <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> literature is made explicit in (<a href="#FSS13">FSS 13</a>).</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Lie+theory">higher Lie theory</a></th><th><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/super+Lie+n-algebra">super Lie n-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> “FDA” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{g})</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> <p>The construction in <a href="#Nieuwenhuizen82">van Nieuwenhuizen 82</a> in turn was motivated by the <a class="existingWikiWord" href="/nlab/show/Sullivan+algebras">Sullivan algebras</a> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> (<a href="#rational+homotopy+theory#Sullivan77">Sullivan 77</a>). Indeed, their dual incarnations in rational homotopy theory are <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> (<a href="#rational+homotopy+theory#Quillen69">Quillen 69</a>), hence a special case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>This close relation between <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/higher+Lie+theory">higher Lie theory</a> might have remained less of a secret had it not been for the focus of <a class="existingWikiWord" href="/nlab/show/Sullivan+minimal+models">Sullivan minimal models</a> on the non-<a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> case, which excludes the ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> from the picture. But the Quillen model of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> effectively says that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/rational+topological+space">rational topological space</a> then its <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega X</annotation></semantics></math> is reflected, infinitesimally, by an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>. This perspective began to receive more attention when the <a class="existingWikiWord" href="/nlab/show/Sullivan+construction">Sullivan construction</a> in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> was concretely identified as higher <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> in <a href="#Lie+integration#Henriques">Henriques 08</a>. A modern review that makes this <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>-theoretic nature of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> manifest is in <a href="BuijsFelixMurillo12">Buijs-Félix-Murillo 12, section 2</a>.</p> </div> <h2 id="definition">Definition</h2> <h3 id="in_terms_of_algebras_over_an_operad">In terms of algebras over an operad</h3> <p>An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra</strong> is an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> over the <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+operad">L-∞ operad</a>.</p> <p>In the following we spell out in detail what this means in components.</p> <h3 id="DefinitionViaHigherBrackets">In terms of higher brackets</h3> <p>We now state the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras that is most directly related to the traditional definition of ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>, namely as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> equipped with <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 <a class="existingWikiWord" href="/nlab/show/multilinear+maps">multilinear</a> and graded-skew symmetric maps <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><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,\cdots,-]</annotation></semantics></math> – the “brackets” – that satisfy a generalization of the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a>.</p> <p>To that end, we here choose grading conventions such that the following definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras reduces to that of ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is concentrated in <em>degree zero</em>. Moreover we take the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> of the underlying <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra to have degree <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> (“homological grading”). Together this means in particular that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <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 \geq 1</annotation></semantics></math>, if it is concentrated in degrees 0 to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n-1</annotation></semantics></math>.</p> <p>Beware that there are also other conventions possible, and there are other conventions in use, for both these choices, leading to different signs in the following formulas.</p> <div class="num_defn" id="GradedSignatureOfPermutation"> <h6 id="definition_2">Definition</h6> <p><strong>(graded signature of a permuation)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo>=</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{v} = (v_1, v_2, \cdots, v_n) </annotation></semantics></math></div> <p>be an <a class="existingWikiWord" href="/nlab/show/n-tuple">n-tuple</a> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> of homogeneous degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\vert v_i \vert \in \mathbb{Z}</annotation></semantics></math>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>V</mi> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo></mrow></msub></mrow><annotation encoding="application/x-tex">v_i \in V_{\vert v_i\vert}</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> elements, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>σ</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{\vert \sigma \vert}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/signature+of+a+permutation">signature of the permutation</a>, which is by definition equal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">(-1)^k</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is the composite of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> permutations that each exchange precisely one pair of neighboring elements.</p> <p>We say that the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{v}</annotation></semantics></math>-graded signature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \chi(\sigma, v_1, \cdots, v_n) \;\in\; \{-1,+1\} </annotation></semantics></math></div> <p>is the product of the <a class="existingWikiWord" href="/nlab/show/signature+of+a+permutation">signature of the permutation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>σ</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{\vert \sigma \vert}</annotation></semantics></math> with a factor of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>j</mi></msub><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{\vert v_i \vert \vert v_j \vert}</annotation></semantics></math> for each interchange of neighbours <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>⋯</mi><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>v</mi> <mi>j</mi></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cdots v_i,v_j, \cdots )</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>⋯</mi><msub><mi>v</mi> <mi>j</mi></msub><mo>,</mo><msub><mi>v</mi> <mi>i</mi></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cdots v_j,v_i, \cdots )</annotation></semantics></math> involved in the decomposition of the permuation as a sequence of swapping neighbour pairs.</p> </div> <div class="num_defn" id="LInfinityDefinitionViaGeneralizedJacobiIdentity"> <h6 id="definition_3">Definition</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra is</p> <ol> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, <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 \geq 1</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/multilinear+map">multilinear map</a>, called the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-ary bracket</em>, of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>⋯</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><munder><munder><mrow><mi>𝔤</mi><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><mi>𝔤</mi></mrow><mo>⏟</mo></munder><mrow><mi>n</mi><mspace width="thickmathspace"></mspace><mtext>copies</mtext></mrow></munder><mo>⟶</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> l_n(\cdots) \;\coloneqq\; [-,-, \cdots, -]_n \;\colon\; \underset{n \; \text{copies}}{\underbrace{\mathfrak{g} \otimes \cdots \otimes \mathfrak{g}}} \longrightarrow \mathfrak{g} </annotation></semantics></math></div> <p>and of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n-2</annotation></semantics></math></p> <p>(if one includes here <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> then one speaks of a <em><a class="existingWikiWord" href="/nlab/show/curved+L-infinity+algebra">curved L-infinity algebra</a></em>)</p> </li> </ol> <p>such that the following conditions hold:</p> <ol> <li> <p>(<strong>graded skew symmetry</strong>) each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">l_n</annotation></semantics></math> is graded antisymmetric, in that for every <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> elements and for every <a class="existingWikiWord" href="/nlab/show/n-tuple">n-tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1, \cdots, v_n)</annotation></semantics></math> of homogeneously graded elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>𝔤</mi> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo></mrow></msub></mrow><annotation encoding="application/x-tex">v_i \in \mathfrak{g}_{\vert v_i \vert}</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>⋅</mo><msub><mi>l</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> l_n(v_{\sigma(1)}, v_{\sigma(2)},\cdots ,v_{\sigma(n)}) = \chi(\sigma,v_1,\cdots, v_n) \cdot l_n(v_1, v_2, \cdots v_n) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi(\sigma,v_1,\cdots, v_n)</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1,\cdots,v_n)</annotation></semantics></math>-graded signature of the permuation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#GradedSignatureOfPermutation"></a>;</p> </li> <li> <p>(<strong>strong homotopy <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a></strong>) for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, and for all <a class="existingWikiWord" href="/nlab/show/n-tuple"><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>-tuples</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1, \cdots, v_n)</annotation></semantics></math> of homogeneously graded elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>𝔤</mi> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">|</mo></mrow></msub></mrow><annotation encoding="application/x-tex">v_i \in \mathfrak{g}_{\vert v_i \vert}</annotation></semantics></math> the following <a class="existingWikiWord" href="/nlab/show/equation">equation</a> holds</p> <div class="maruku-equation" id="eq:LInfinityJacobiIdentity"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>ℕ</mi></mrow></mrow><mrow><mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>=</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mrow></mfrac></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>UnShuff</mi><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><mi>χ</mi><mo stretchy="false">(</mo><mi>σ</mi><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msup><msub><mi>l</mi> <mi>j</mi></msub><mrow><mo>(</mo><msub><mi>l</mi> <mi>i</mi></msub><mrow><mo>(</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \sum_{{i,j \in \mathbb{N}} \atop {i+j = n+1}} \sum_{\sigma \in UnShuff(i,j-1)} \chi(\sigma,v_1, \cdots, v_{n}) (-1)^{i(j-1)} l_{j} \left( l_i \left( v_{\sigma(1)}, \cdots, v_{\sigma(i)} \right), v_{\sigma(i+1)} , \cdots , v_{\sigma(n)} \right) = 0 \,, </annotation></semantics></math></div> <p>where the inner sum runs over all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i,j-1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unshuffles">unshuffles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> is the graded signature sign from def. <a class="maruku-ref" href="#GradedSignatureOfPermutation"></a>.</p> </li> </ol> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>In lowest degrees the generalized Jacobi identity says that</p> <ol> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math>: the unary map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>≔</mo><msub><mi>l</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\partial \coloneqq l_1</annotation></semantics></math> squares to 0:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo stretchy="false">(</mo><mo>∂</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \partial (\partial(v_1)) = 0 </annotation></semantics></math></div></li> <li> <p>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 unary map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo></mrow><annotation encoding="application/x-tex">\partial</annotation></semantics></math> is a graded <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of the binary map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">|</mo></mrow></msup><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo>∂</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> - [\partial v_1, v_2] - (-1)^{\vert v_1 \vert \vert v_2 \vert} [\partial v_2, v_1] + \partial [v_1, v_2] = 0 </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><mo>∂</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">|</mo></mrow></msup><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo>∂</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial [v_1, v_2] = [\partial v_1, v_2] + (-1)^{\vert v_1 \vert} [v_1, \partial v_2] \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>When all higher brackets vanish, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mrow><mi>k</mi><mo>></mo><mn>2</mn></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">l_{k \gt 2}= 0</annotation></semantics></math> then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> [[v_1,v_2],v_3] + (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)} [[v_2,v_3],v_1] + (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)} [[v_1,v_3],v_2] = 0 </annotation></semantics></math></div> <p>this is the graded <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a>. So in this case the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra is equivalently a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">l_3</annotation></semantics></math> is possibly non-vanishing, then on elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math> on which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mo>=</mo><msub><mi>l</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\partial = l_1</annotation></semantics></math> vanishes, the generalized Jacobi identity for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math> gives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">|</mo><mo>+</mo><mo stretchy="false">|</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∂</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [[v_1,v_2],v_3] + (-1)^{\vert v_1 \vert (\vert v_2 \vert + \vert v_3 \vert)} [[v_2,v_3],v_1] + (-1)^{\vert v_2 \vert (\vert v_1 \vert + \vert v_3 \vert)} [[v_1,v_3],v_2] = - \partial [v_1, v_2, v_3] \,. </annotation></semantics></math></div> <p>This shows that the Jacobi identity holds up to an “exact” term, hence up to homotopy.</p> </div> <h3 id="ReformulationInSemifreeDgCoalgebra">In terms of semifree differential coalgebra</h3> <p>In (<a href="#LadaStasheff92">Lada-Stasheff 92</a>) it was pointed out that the higher brackets of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra (def. <a class="maruku-ref" href="#LInfinityDefinitionViaGeneralizedJacobiIdentity"></a>) induce on the graded-co-commutative <a class="existingWikiWord" href="/nlab/show/cofree+coalgebra">cofree coalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\vee^\bullet \mathfrak{g}</annotation></semantics></math> over the underlying graded vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> the structure of a <a class="existingWikiWord" href="/nlab/show/differential+graded+coalgebra">differential graded coalgebra</a>, with differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</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>+</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">D = [-] + [-,-] + [-,-,-] + \cdots</annotation></semantics></math> the sum of the higher brackets, extended as graded <a class="existingWikiWord" href="/nlab/show/coderivations">coderivations</a>. The higher Jacobi identity is equivalently the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">D^2 = 0</annotation></semantics></math>. In (<a href="#LadaMarkl94">Lada-Markl 94</a>) it was observed that conversely, such “semifree” <a class="existingWikiWord" href="/nlab/show/differential+graded+coalgebras">differential graded coalgebras</a> are an equivalent incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>(If one uses unital dg-co-algebras then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algbras encoded with way are generally <a class="existingWikiWord" href="/nlab/show/curved+L-infinity+algebras">curved L-infinity algebras</a>. To restrict to the non-curved one one either considers co-augmented unital dg-co-algebras or non-unital coalgebras.)</p> <p>Notice that this immediately imples that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is degreewise finite dimensional, then passing to <a class="existingWikiWord" href="/nlab/show/dual+vector+spaces">dual vector spaces</a> turns semifree <a class="existingWikiWord" href="/nlab/show/differential+graded+coalgebra">differential graded coalgebra</a> into <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree</a> <a class="existingWikiWord" href="/nlab/show/differential+graded+algebras">differential graded algebras</a>, which hence are <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a>-equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> an ordinary finite dimensional <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, then this dg-algebras is its <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, hence we may generally speak of <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> (and also more generally, if one invokes <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a>, see at <a href="model+structure+for+L-infinity+algebras#OnProAlg">model structure for L-infinity algebras – Use of pro-dg-algebras</a> ).</p> <p>In term of the <a class="existingWikiWord" href="/nlab/show/operad">operadic</a> definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras <a href="#InTermsOfAlgebrasOverAnOperad">above</a> this equivalence is an incarnation of the <a class="existingWikiWord" href="/nlab/show/Koszul+duality">Koszul duality</a> between the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a> and the <a class="existingWikiWord" href="/nlab/show/commutative+operad">commutative operad</a>.</p> <p>We now spell out this dg-coalgebraic incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>A (connected) <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra</strong> is</p> <ul> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\mathbb{N}_+</annotation></semantics></math>-graded vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>;</p> </li> <li> <p>equipped with a differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>:</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi><mo>→</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g}</annotation></semantics></math> of degree <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> on the <a class="existingWikiWord" href="/nlab/show/free+graded+co-commutative+coalgebra">free graded co-commutative coalgebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> that squares to 0</p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D^2 = 0 \,. </annotation></semantics></math></div> <p>Here the free graded co-commutative co-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\vee^\bullet \mathfrak{g}</annotation></semantics></math> is, as a vector space, the same as the graded <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\wedge^\bullet \mathfrak{g}</annotation></semantics></math> whose elements we write as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>3</mn><msub><mi>t</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>t</mi> <mn>3</mn></msub><mo>+</mo><msub><mi>t</mi> <mn>3</mn></msub><mo>∨</mo><msub><mi>t</mi> <mn>4</mn></msub><mo>∨</mo><msub><mi>t</mi> <mn>5</mn></msub></mrow><annotation encoding="application/x-tex"> 3 t_1 \vee t_2 + t_3 + t_3 \vee t_4 \vee t_5 </annotation></semantics></math></div> <p>etc (where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∨</mo></mrow><annotation encoding="application/x-tex">\vee</annotation></semantics></math> is just a funny way to write the wedge <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∧</mo></mrow><annotation encoding="application/x-tex">\wedge</annotation></semantics></math>, in order to remind us that:…)</p> <p>but thought of as equipped with the standard coproduct</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>v</mi> <mn>2</mn></msub><mi>⋯</mi><mo>∨</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∝</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo>±</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>∨</mo><mi>⋯</mi><mo>∨</mo><msub><mi>v</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∨</mo><mi>⋯</mi><mo>∨</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Delta (v_1 \vee v_2 \cdots \vee v_n) \propto \sum_i \pm (v_1 \vee \cdots \vee v_i) \otimes (v_{i+1} \vee \cdots \vee v_n) </annotation></semantics></math></div> <blockquote> <p>(work out or see the references for the signs and prefacors).</p> </blockquote> <p>Since this is a <em>free</em> graded co-commutative coalgebra, one can see that any differential</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>:</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi><mo>→</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> D : \vee^\bullet \mathfrak{g} \to \vee^\bullet \mathfrak{g} </annotation></semantics></math></div> <p>on it is fixed by its value “on cogenerators” (a statement that is maybe unfamiliar, but simply the straightforward dual of the more familar statement to which we come below, that differentials on free graded algebras are fixed by their action on generators) which means that we can decompose <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> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>D</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>D</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>D</mi> <mn>3</mn></msub><mo>+</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> D = D_1 + D_2 + D_3 + \cdots \,, </annotation></semantics></math></div> <p>where each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">D_i</annotation></semantics></math> acts as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>l</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">l_i</annotation></semantics></math> when evaluated on a homogeneous element of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mn>1</mn></msub><mo>∨</mo><mi>⋯</mi><mo>∨</mo><msub><mi>t</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t_1 \vee \cdots \vee t_n</annotation></semantics></math> and is then uniquely extended to all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\vee^\bullet \mathfrak{g}</annotation></semantics></math> by extending it as a <em>coderivation</em> on a coalgebra.</p> <p>For instance <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">D_2</annotation></semantics></math> acts on homogeneous elements of word lenght 3 as</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> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>∨</mo><msub><mi>t</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∨</mo><msub><mi>t</mi> <mn>3</mn></msub><mo>±</mo><mi>permutations</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D_2(t_1 \vee t_2 \vee t_3) = D_2(t_1, t_2)\vee t_3 \pm permutations \,. </annotation></semantics></math></div> <blockquote> <p>exercise for the reader: spell this all out more in detail with all the signs and everyrthing. Possibly by looking it up in the references given below.</p> </blockquote> <p>Using this, one checks that the simple condition that <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> squares to 0 is precisely equivalent to the infinite tower of generalized Jacobi identities:</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><msup><mi>D</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>⇔</mo><mrow><mo>(</mo><mo>∀</mo><mi>n</mi><mo>:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>=</mo><mi>n</mi></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>shuffles</mi><mi>σ</mi></mrow></munder><mo>±</mo><msub><mi>l</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>l</mi> <mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (D^2 = 0) \Leftrightarrow \left( \forall n : \sum_{i+j = n} \sum_{shuffles \sigma} \pm l_i (l_j (v_{\sigma(1)}, \cdots, v_{\sigma(j)} ) , v_{\sigma(j+1)} , \cdots , v_{\sigma(n)} ) = 0 \right) \,. </annotation></semantics></math></div> <p>So in conclusion we have:</p> <div class="standout"> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra is a <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a> whose underlying <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> is cofree and concentrated in negative degree.</p> </div> <h3 id="ReformulationInTermsOfSemifreeDGAlgebra">In terms of semifree differential algebra</h3> <p>The reformulation of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra as simply a semi-co-free graded-co-commutative coalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∨</mo> <mo>•</mo></msup><mi>𝔤</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\vee^\bullet \mathfrak{g}, D)</annotation></semantics></math> is a useful repackaging of the original definition, but the coalgebraic aspect tends to be not only unfamiliar, but also a bit inconvenient. At least when the graded vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is degreewise <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a>, we may simply pass to its degreewise dual graded vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math>.</p> <p>(Fully generally the following works when using not just dg-algebras but <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> in <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a>, see at <em><a href="model+structure+for+L-infinity+algebras#OnProAlg">model structure for L-infinity algebras – Use of pro-dg-algebras</a></em>).</p> <p>Its <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math> is then naturally equipped with an ordinary differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><msup><mi>D</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">d = D^*</annotation></semantics></math> which acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\omega \in \wedge^\bullet \mathfrak{g}^*</annotation></semantics></math> as</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>d</mi><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>∨</mo><mi>⋯</mi><mo>∨</mo><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo>±</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>∨</mo><mi>⋯</mi><mo>∨</mo><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (d \omega) (t_1 \vee \cdots \vee t_n) = \pm \omega(D(t_1 \vee \cdots \vee t_n)) \,. </annotation></semantics></math></div> <p>When the grading-dust has settled one finds that with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>=</mo><mi>k</mi><mo>⊕</mo><msubsup><mi>𝔤</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊕</mo><mo stretchy="false">(</mo><msubsup><mi>𝔤</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>∧</mo><msubsup><mi>𝔤</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>⊕</mo><msubsup><mi>𝔤</mi> <mn>2</mn> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mo>⊕</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \wedge^\bullet \mathfrak{g}^* = k \oplus \mathfrak{g}^*_1 \oplus (\mathfrak{g}^*_1 \wedge \mathfrak{g}^*_1 \oplus \mathfrak{g}^*_2) \oplus \cdots </annotation></semantics></math></div> <p>with the ground field in degree 0, the degree 1-elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> in degree 1, etc, that <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> is of degree +1 and of course squares to 0</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d^2 = 0 \,. </annotation></semantics></math></div> <p>This means that we have a <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a></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>𝔤</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CE(\mathfrak{g}) := (\wedge^\bullet \mathfrak{g}^*, d) \,. </annotation></semantics></math></div> <p>In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> happens to be an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, this is the ordinary <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of this Lie algebra. Hence we should generally call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{g})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math>.</p> <p>One observes that this construction is bijective: every (degreewise finite dimensional) cochain <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a> generated in positive degree comes from a (degreewise finite dimensional) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra this way.</p> <p>This means that we may just as well <em>define</em> a (degreewise finite dimensional) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra as an object in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of (degreewise finite dimensional) commutative <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s that are <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a>s and generated in positive degree.</p> <p>(In general this corresponds to <a class="existingWikiWord" href="/nlab/show/curved+L-infinity+algebra">curved L-infinity algebra</a>. The flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> dually correspond to the dg-algebras which are <a class="existingWikiWord" href="/nlab/show/augmented+algebra">augmented</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>, i.e for which the canonical projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">CE(\mathfrak{g}) \longrightarrow \mathbb{R}</annotation></semantics></math> is a homomorphism of dg-algebras.)</p> <p>And this turns out to be one of the most useful perspectives on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>In particular, if we simply drop the condition that the dg-algebra be generated in positive degree and allow it to be generated in non-negative degree over the algebra in degree 0, then we have the notion of the (Chevalley-Eilenberg algebra of) an <a class="existingWikiWord" href="/nlab/show/L-infinity-algebroid">L-infinity-algebroid</a>.</p> <h4 id="DGAlgebraDetails">Details</h4> <p>We discuss in explit detail the computation that shows that an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\wedge^\bullet \mathfrak{g}^*</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> be a degreewise finite-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\mathbb{N}_+</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> equipped with multilinear graded-symmetric maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo>:</mo><msup><mi>Sym</mi> <mi>k</mi></msup><mi>𝔤</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> [-,\cdots,-]_k : Sym^k \mathfrak{g} \to \mathfrak{g} </annotation></semantics></math></div> <p>of degree -1, for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}_+</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>t</mi> <mi>a</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t_a\}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/basis">basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>t</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t^a\}</annotation></semantics></math> a dual basis of the degreewise dual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math>. Equip the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sym</mi> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">Sym^\bullet \mathfrak{g}^*</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>Sym</mi> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>Sym</mi> <mo>•</mo></msup><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> d : Sym^\bullet \mathfrak{g}^* \to Sym^\bullet \mathfrak{g}^* </annotation></semantics></math></div> <p>defined on generators by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>:</mo><msup><mi>t</mi> <mi>a</mi></msup><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">[</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msub><msubsup><mo stretchy="false">]</mo> <mi>k</mi> <mi>a</mi></msubsup><mspace width="thinmathspace"></mspace><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d : t^a \mapsto - \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, \cdots, t_{a_k}]^a_k \, t^{a_1} \wedge \cdots \wedge t^{a_k} \,. </annotation></semantics></math></div> <p>Here we take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>t</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">t^a</annotation></semantics></math> to be of the same degree as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">t_a</annotation></semantics></math>. Therefore this derivation has degree +1.</p> <p>We compute the square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mo>=</mo><mi>d</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">d^2 = d \circ d</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><mi>d</mi><mi>d</mi><msup><mi>t</mi> <mi>a</mi></msup></mtd> <mtd><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">)</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><mfrac><mn>1</mn><mrow><mi>k</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">[</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msub><msubsup><mo stretchy="false">]</mo> <mi>k</mi> <mi>a</mi></msubsup><mspace width="thinmathspace"></mspace><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mi>l</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>t</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>b</mi> <mi>l</mi></msub></mrow></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msub><msup><mo stretchy="false">]</mo> <mi>a</mi></msup><mspace width="thinmathspace"></mspace><msup><mi>t</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>b</mi> <mi>l</mi></msub></mrow></msup><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d d t^a &= d (-1)\sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, \cdots, t_{a_k}]^a_k \, t^{a_1} \wedge \cdots \wedge t^{a_k} \\ & = \sum_{k,l = 1}^\infty \frac{1}{(k-1)! l!} [[t_{b_1}, \cdots, t_{b_l}], t_{a_2}, \cdots, t_{a_k}]^a \, t^{b_1} \wedge \cdots \wedge t^{b_l} \wedge t^{a_2} \wedge \cdots \wedge t^{a_{k}} \end{aligned} \,. </annotation></semantics></math></div> <p>Here the wedge product on the right projects the nested bracket onto its graded-symmetric components. This is produced by summing over all <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>Σ</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma \in \Sigma_{k+l-1}</annotation></semantics></math> weighted by the Koszul-<a class="existingWikiWord" href="/nlab/show/signature">signature</a> of the permutation:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><msub><mi>Σ</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></msup><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mi>l</mi><mo>!</mo></mrow></mfrac><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>t</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>b</mi> <mi>l</mi></msub></mrow></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msub><msup><mo stretchy="false">]</mo> <mi>a</mi></msup><mspace width="thinmathspace"></mspace><msup><mi>t</mi> <mrow><msub><mi>b</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>b</mi> <mi>l</mi></msub></mrow></msup><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mi>k</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = \sum_{k,l = 1}^\infty \frac{1}{(k+l-1)!} \sum_{\sigma \in \Sigma_{k+l-1}} (-1)^{sgn(\sigma)} \frac{1}{(k-1)! l!} [[t_{b_1}, \cdots, t_{b_l}], t_{a_2}, \cdots, t_{a_k}]^a \, t^{b_1} \wedge \cdots \wedge t^{b_l} \wedge t^{a_2} \wedge \cdots \wedge t^{a_{k}} \,. </annotation></semantics></math></div> <p>The sum over all permutations decomposes into a sum over the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>l</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(l,k-1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unshuffle">unshuffle</a>s and a sum over permutations that act inside the first <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> and the last <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k-1)</annotation></semantics></math> indices. By the graded-symmetry of the bracket, the latter do not change the value of the nested bracket. Since there are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo><mi>l</mi><mo>!</mo></mrow><annotation encoding="application/x-tex">(k-1)! l!</annotation></semantics></math> many of them, we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>,</mo><mi>l</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><mfrac><mn>1</mn><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>Unsh</mi><mo stretchy="false">(</mo><mi>l</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo>[</mo><mrow><mo>[</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>l</mi></msub></mrow></msub><mo>]</mo></mrow><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mrow><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>t</mi> <mrow><msub><mi>a</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = \sum_{k,l = 1}^\infty \frac{1}{(k+l-1)!} \sum_{\sigma \in Unsh(l,k-1)} (-1)^{sgn(\sigma)} \left[\left[t_{a_1}, \cdots, t_{a_l}\right], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}\right] \, t^{a_1} \wedge \cdots \wedge t^{a_{k+l-1}} \,. </annotation></semantics></math></div> <p>Therefore the condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>d</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d^2 = 0</annotation></semantics></math> is equivalent to the condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>σ</mi><mo>∈</mo><mi>Unsh</mi><mo stretchy="false">(</mo><mi>l</mi><mo>,</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>sgn</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></msup><mrow><mo>[</mo><mrow><mo>[</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>l</mi></msub></mrow></msub><mo>]</mo></mrow><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mrow><mi>l</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>]</mo></mrow><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \sum_{k+l = n-1} \sum_{\sigma \in Unsh(l,k-1)} (-1)^{sgn(\sigma)} \left[\left[t_{a_1}, \cdots, t_{a_l}\right], t_{a_{l+1}}, \cdots, t_{a_{k+l-1}}\right] = 0 </annotation></semantics></math></div> <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> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>t</mi> <mrow><msub><mi>a</mi> <mi>i</mi></msub></mrow></msub><mo>∈</mo><mi>𝔤</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{t_{a_i} \in \mathfrak{g}\}</annotation></semantics></math>. This is equation <a class="maruku-eqref" href="#eq:LInfinityJacobiIdentity">(1)</a> which says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>𝔤</mi><mo>,</mo><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>…</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\mathfrak{g}, \{[-,\dots,-]_k\}\}</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra.</p> <h3 id="InTermsOfAlgebrasOverAnOperad">In terms of algebras over an operad</h3> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras are precisely the <a class="existingWikiWord" href="/nlab/show/algebras+over+an+operad">algebras over an operad</a> of the cofibrant resolution of the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a>.</p> <h2 id="examples">Examples</h2> <h3 id="special_cases">Special cases</h3> <ul> <li> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is concentrated in the first <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> degree is a <strong>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>-algebra</strong> (sometimes also: “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">L_n</annotation></semantics></math>-algebra”).</p> </li> <li> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra for which only the unary operation and the binary bracket are non-trivial is a <strong><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a></strong>: a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a>. From the point of view of higher Lie theory this is a <strong>strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra</strong>: one for which the Jacobi identity does happen to hold “on the nose”, not just up to nontrivial coherent isomorphisms.</p> </li> <li> <p>So in particular</p> <ul> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra generated just in degree 1 is an ordinary <strong><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></strong> ;</p> </li> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra generated just in degree 1 and 2 is a <strong><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></strong> ;</p> <ul> <li>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra generated just in degree 1 and 2 and with at most binary brackets is a <strong><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></strong> , equivalently encoded in a <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a>.</li> </ul> </li> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra generated just in degree 1, 2 and 3 is a <strong><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></strong> ;</p> </li> </ul> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> is a Lie algebra over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{K}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">b^{k-1}\mathbb{K}</annotation></semantics></math> is the complex consisting of the field <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> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>−</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1-k</annotation></semantics></math>, then an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>b</mi> <mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>𝕂</mi></mrow><annotation encoding="application/x-tex">b^{k-1}\mathbb{K}</annotation></semantics></math> is precisely a degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cocycle</a>.</p> </li> <li> <p>The skew-symmetry of the Lie bracket is retained strictly in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras. It is expected that weakening this, too, yields a more general <a class="existingWikiWord" href="/nlab/show/vertical+categorification">vertical categorification</a> of Lie algebras. 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> this has been worked out by Dmitry Roytenberg: <a href="http://arxiv.org/abs/0712.3461">On weak Lie 2-algebras</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Lie+infinity-algebroid">algebroid</a>s.</p> </li> <li> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra with only <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">D_n</annotation></semantics></math> non-vanishing is called an <strong><a class="existingWikiWord" href="/nlab/show/n-Lie+algebra">n-Lie algebra</a></strong> – to be distinguished from a <em>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>-algebra</em> ! However, in large parts of the literature <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>-Lie algebras are considered for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">D_n</annotation></semantics></math> is <em>not</em> of the required homogeneous degree in the grading, or in which no grading is considered in the first place. Such <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>-Lie algebras are not special examples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras, then. For more see <a class="existingWikiWord" href="/nlab/show/n-Lie+algebra">n-Lie algebra</a>.</p> </li> <li> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra internal to <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>s is a <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a>.</p> </li> </ul> <h3 id="classes_of_examples">Classes of examples</h3> <ul> <li> <p><span class="newWikiWord">automorphism Lie 2-algebra<a href="/nlab/new/automorphism+Lie+2-algebra">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+derivation+Lie+2-algebra">inner derivation Lie 2-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>For every <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 algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> there is its <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a>. In terms of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> this models the rational automorphism group of the <a class="existingWikiWord" href="/nlab/show/rational+space">rational space</a> corresponding 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">\mathfrak{g}</annotation></semantics></math>.</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+n-algebra">Heisenberg Lie n-algebra</a> of an <a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a> or more generally of an <a class="existingWikiWord" href="/nlab/show/n-plectic+smooth+infinity-groupoid">n-plectic smooth infinity-groupoid</a></p> </li> <li> <p>some classes of <a class="existingWikiWord" href="/nlab/show/W-algebras">W-algebras</a> are claimed to induce <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras in <a href="W+algebra#BlumenhagenFuchsTraube17">Blumenhagen-Fuchs-Traube 17</a></p> </li> </ul> <h3 id="specific_examples">Specific examples</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-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> </ul> <h2 id="properties">Properties</h2> <h3 id="IndConilpotency">Ind-Conilpotency</h3> <div class="num_remark" id="IndConilpotency"> <h6 id="remark_2">Remark</h6> <p><strong>(<a href="#Pridham">Pridham 10, remark 3.15, remark 3.13</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra, then its CE chain <a class="existingWikiWord" href="/nlab/show/dgc-coalgebra">dgc-coalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CE</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE_\bullet(\mathfrak{g})</annotation></semantics></math> (<a href="#ReformulationInSemifreeDgCoalgebra">above</a>) is <em>ind-conilpotent</em>.</p> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CE</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE_\bullet(\mathfrak{g})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of sub-dg-coalgebras which are conilpotent, in that for each of them there is <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> such that their <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>-fold coproduct vanishes. As such these are like “co-<a class="existingWikiWord" href="/nlab/show/local+Artin+algebras">local Artin algebras</a>”.</p> <p>Moreover, since every dg-coalgebra is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of its finite-dimensonal subalgebras (see at <em><a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a></em> the section <em><a href="differential+graded+coalgebra#AsFilteredColimits">As filtered colimits of finite-dimensional pieces</a></em>), this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CE</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE_\bullet(\mathfrak{g})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of finite dimensional conilpotent coalgebras.</p> <p>This implies that the dual Chevalley-Eilenberg cochain algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CE</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE^\bullet(\mathfrak{g})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/filtered+limit">filtered limit</a> of finite-dimensional nilpotent <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a> (actual <a class="existingWikiWord" href="/nlab/show/local+Artin+algebras">local Artin algebras</a>).</p> </div> <h3 id="model_category_structure">Model category structure</h3> <p>See <a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>.</p> <h3 id="relation_to_dglie_algebras">Relation to dg-Lie algebras</h3> <p>Every <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> is in an evident way an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra. Dg-Lie algebras are precisely those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras for which 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>-ary brackets 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 \gt 2</annotation></semantics></math> are trivial. These may be thought of as the <em>strict</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras: those for which the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> holds on the nose and all its possible higher coherences are trivial.</p> <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>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0 and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">L_\infty Alg_k</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mn>∞</mn></mrow><annotation encoding="application/x-tex">L\infty</annotation></semantics></math>-algebras over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> <p>Then every object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">L_\infty Alg_k</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>.</p> <p>Moreover, one can find a functorial replacement: there is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></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>:</mo><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> W : L_\infty Alg_k \to L_\infty Alg_k </annotation></semantics></math></div> <p>such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>∈</mo><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g} \in L_\infty Alg_k</annotation></semantics></math></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔨</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{k})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>;</p> </li> <li> <p>there is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>W</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{g} \stackrel{\simeq}{\to} W(\mathfrak{g}) \,. </annotation></semantics></math></div></li> </ol> </div> <p>This appears for instance as (<a href="#KrizMay95">Kriz & May 1995, Cor. 1.6</a>).</p> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/relation+between+L-%E2%88%9E+algebras+and+dg-Lie+algebras">relation between L-∞ algebras and dg-Lie algebras</a></em>.</p> <h3 id="relation_to_lie_groupoids">Relation 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>-Lie groupoids</h3> <p>In generalization to how a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> integrates to a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras integrate to <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a>s.</p> <p>See</p> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></p> <p>and</p> <p><a href="http://ncatlab.org/nlab/show/Lie+infinity-groupoid#LieIntegrated">Lie integrated ∞-Lie groupoids</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> </li> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homomorphism+of+L-%E2%88%9E+algebras">homomorphism of L-∞ algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">infinity-Lie algebra cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+enveloping+E-n+algebra">universal enveloping E-n algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nilpotent+L-%E2%88%9E+algebra">nilpotent L-∞ algebra</a></p> </li> <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/derived+L-%E2%88%9E+algebroid">derived L-∞ algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+of+L-%E2%88%9E+algebras">sheaf of L-∞ algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+L-%E2%88%9E+algebra">quantum L-∞ algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+L-%E2%88%9E+algebras+and+dg-Lie+algebras">relation between L-∞ algebras and dg-Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/EL-%E2%88%9E+algebra">EL-∞ algebra</a></p> </li> </ul> <h2 id="References">References</h2> <div> <h3 id="general">General</h3> <p>The concept of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> as <a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a> equipped with <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 satisfying a generalized <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> is due to:</p> <ul> <li id="Stasheff92"> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras</em>, in <em>Quantum groups</em> Number 1510 in Lecture Notes in Math. Springer, Berlin, 1992 (<a href="https://doi.org/10.1007/BFb0101184">doi:10.1007/BFb0101184</a>).</p> </li> <li id="LadaStasheff92"> <p><a class="existingWikiWord" href="/nlab/show/Tom+Lada">Tom Lada</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Introduction to sh Lie algebras for physicists</em>, Int. J. Theo. Phys. <strong>32</strong> (1993) 1087-1103 [<a href="https://doi.org/10.1007/BF00671791">doi:10.1007/BF00671791</a>, <a href="http://arxiv.org/abs/hep-th/9209099">arXiv:hep-th/9209099</a>]</p> </li> <li id="LadaMarkl94"> <p><a class="existingWikiWord" href="/nlab/show/Tom+Lada">Tom Lada</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Markl">Martin Markl</a>, <em>Strongly homotopy Lie algebras</em>, Communications in Algebra <strong>23</strong> 6 (1995) [<a href="https://doi.org/10.1080/00927879508825335">doi:10.1080/00927879508825335</a>, <a href="http://arxiv.org/abs/hep-th/9406095">arXiv:hep-th/9406095</a>]</p> </li> <li id="Kontsevich97"> <p><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, Section 4.3 of: <em>Deformation quantization of Poisson manifolds, I</em>, Lett. Math. Phys. <strong>66</strong> (2003) 157-216 (<a href="https://arxiv.org/abs/q-alg/9709040">arXiv:q-alg/9709040</a>, <a href="https://doi.org/10.1023/B:MATH.0000027508.00421.bf">doi:10.1023/B:MATH.0000027508.00421.bf</a>)</p> </li> </ul> <p>At least <a href="#Stasheff92">Stasheff 92</a> was following <a href="#Zwiebach92">Zwiebach 92</a>, who had observed that the <a class="existingWikiWord" href="/nlab/show/n-point+functions">n-point functions</a> in <a class="existingWikiWord" href="/nlab/show/closed+string+field+theory">closed string field theory</a> equip the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra structure (see further reference <a href="string+field+theory#ReferencesHomotopyAlgebra">there</a>). Zwiebach, in turn, was following the <a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> due to <a href="#BatalinVilkovisky83">Batalin-Vilkovisky 83</a>, <a href="#BatakinFradkin83">Batakin-Fradkin 83</a>.</p> <p>See also at <em><a href="L-infinity-algebra#History">L-infinity algebra – History</a></em>.</p> <p>Discussion in terms of <a class="existingWikiWord" href="/nlab/show/cofibrant+resolutions">cofibrant resolutions</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a>:</p> <ul> <li id="KrizMay95"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, p. 28 of: <em>Operads, algebras, modules and motives</em>, Astérisque <strong>233</strong>, Société Mathématique de France (1995) (<a href="http://www.math.uchicago.edu/~may/PAPERS/kmbooklatex.pdf">pdf</a>, <a href="http://www.numdam.org/item/?id=AST_1995__233__1_0">numdam:AST_1995__233__1_0</a>)</p> </li> <li id="LodayVallette12"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Loday">Jean-Louis Loday</a>, <a class="existingWikiWord" href="/nlab/show/Bruno+Vallette">Bruno Vallette</a>, Sec. 3.2.12 and onwards in: <em>Algebraic Operads</em>, Grundlehren der mathematischen Wissenschaften <strong>346</strong>, Springer 2012 (<a href="https://www.springer.com/gp/book/9783642303616">ISBN 978-3-642-30362-3</a>, <a href="http://irma.math.unistra.fr/~loday/PAPERS/LodayVallette.pdf">pdf</a>)</p> </li> </ul> <p>A historical survey is</p> <ul> <li id="Stasheff16"><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Higher homotopy structures, then and now</em>, talk at <em><a href="https://www.mpim-bonn.mpg.de/node/6356">Opening workshop</a></em> of <em><a href="https://www.mpim-bonn.mpg.de/node/5883">Higher Structures in Geometry and Physics</a></em>, MPI Bonn 2016 (<a class="existingWikiWord" href="/nlab/files/StasheffHomotopyStructuresReview.pdf" title="pdf">pdf</a>, <a href="https://arxiv.org/abs/1809.02526">arXiv:1809.02526</a>)</li> </ul> <p>See also</p> <ul> <li id="Daily04"> <p>Marilyn Daily, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-structures</em>, PhD thesis, 2004 (<a href="http://www.lib.ncsu.edu/resolver/1840.16/5282">web</a>)</p> </li> <li> <p>Klaus Bering, <a class="existingWikiWord" href="/nlab/show/Tom+Lada">Tom Lada</a>, <em>Examples of Homotopy Lie Algebras</em> Archivum Mathematicum (<a href="http://arxiv.org/abs/0903.5433">arXiv:0903.5433</a>)</p> </li> </ul> <p>Comprehensive survey with emphasis on <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra+cohomology"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebra cohomology</a>:</p> <ul> <li>Ben Reinhold, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras and their cohomology</em>, Emergent Scientist <strong>3</strong> 4 (2019) &lbrack;<a href="https://doi.org/10.1051/emsci/2019003">doi:10.1051/emsci/2019003</a>&rbrack;</li> </ul> <p>Review for the special case of <a class="existingWikiWord" href="/nlab/show/Lie+2-algebras">Lie 2-algebras</a> with emphasis on the perspective of <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Alissa+Crans">Alissa Crans</a>, <em>Higher-dimensional algebra VI: Lie 2-algebras</em>, <a href="http://www.tac.mta.ca/tac/volumes/12/14/12-14abs.html">TAC</a> 12, (2004), 492–528. (<a href="http://arxiv.org/abs/math/0307263">arXiv:math/0307263</a>)</li> </ul> <h3 id="as_models_for_rational_homotopy_types">As models for rational homotopy types</h3> <p>That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras are models for <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> is implicit in <a href="rational+homotopy+theory#Quillen69">Quillen 69</a> (via their <a href="model+structure+on+dg-Lie+algebras#RectificationResolution">equivalence with dg-Lie algebras</a>) and was made explicit in <a href="#Hinich98">Hinich 98</a>. Exposition is in</p> <ul> <li id="BuijsFelixMurillo12"><a class="existingWikiWord" href="/nlab/show/Urtzi+Buijs">Urtzi Buijs</a>, <a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, <a class="existingWikiWord" href="/nlab/show/Aniceto+Murillo">Aniceto Murillo</a>, section 2 of <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-rational homotopy of mapping spaces</em> (<a href="https://arxiv.org/abs/1209.4756">arXiv:1209.4756</a>), published as <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-models of based mapping spaces</em>, J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524.</li> </ul> <p>and genralization to non-<a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> rational spaces is discussed in</p> <ul> <li id="BuijsMurillo12"><a class="existingWikiWord" href="/nlab/show/Urtzi+Buijs">Urtzi Buijs</a>, <a class="existingWikiWord" href="/nlab/show/Aniceto+Murillo">Aniceto Murillo</a>, <em>Algebraic models of non-connected spaces and homotopy theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras</em>, Advances in Mathematics 236 (2013): 60-91. (<a href="https://arxiv.org/abs/1204.4999">arXiv:1204.4999</a>)</li> </ul> <h3 id="ReferencesInPhysics"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras in physics</h3> <p>The following lists, mainly in chronological order of their discovery, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> structures appearing in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, notably in <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>, <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, higher <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>/<a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-models">AKSZ sigma-models</a> and <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a>.</p> <p>For more see also at <em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></em>.</p> <h4 id="in_supergravity">In supergravity</h4> <p>Implicitly, in their equivalent <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> guise of <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> (see <a href="#ReformulationInTermsOfSemifreeDGAlgebra">above</a>), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> – in fact <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebras">super L-∞ algebras</a> – play a pivotal role in the <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formulation of supergravity</a> at least since</p> <ul> <li id="Nieuwenhuizen82"> <p><a class="existingWikiWord" href="/nlab/show/Peter+van+Nieuwenhuizen">Peter van Nieuwenhuizen</a>, <em>Free Graded Differential Superalgebras</em>, in: <em>Group Theoretical Methods in Physics</em>, Lecture Notes in Physics <strong>180</strong>, Springer (1983) 228–247 [<a href="https://doi.org/10.1007/3-540-12291-5_29">doi:10.1007/3-540-12291-5_29</a>, <a href="http://inspirehep.net/record/182644">spire:182644</a>]</p> </li> <li id="DAuriaFre"> <p><a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, <em><a class="existingWikiWord" href="/nlab/files/GeometricSupergravity.pdf" title="Geometric Supergravity in D=11 and its hidden supergroup">Geometric Supergravity in D=11 and its hidden supergroup</a></em>, Nuclear Physics B <strong>201</strong> (1982) 101-140 [<a href="https://doi.org/10.1016/0550-3213(82)90376-5">doi:10.1016/0550-3213(82)90376-5</a>, <a class="existingWikiWord" href="/nlab/files/GeometricSupergravityErrata.pdf" title="errata">errata</a>]</p> </li> <li id="CastellaniDAuriaFre"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, Ch III.6 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>, ch III.6: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChIII6.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, §6.3 in: <em>Gravity, a Geometrical Course</em>, Volume 2: <em>Black Holes, Cosmology and Introduction to Supergravity</em>, Springer (2013) [<a href="https://doi.org/10.1007/978-94-007-5443-0">doi:10.1007/978-94-007-5443-0</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, §6 in: <em>Supergravity in the group-geometric framework: a primer</em>, Fortschr. Phys. <strong>66</strong> 4 (2018) [<a href="https://doi.org/10.1002/prop.201800014">doi:10.1002/prop.201800014</a>, <a href="https://arxiv.org/abs/1802.03407">arXiv:1802.03407</a>]</p> </li> </ul> <p>where they are called “free differential algebras” (“FDA”s, apparently following <a href="D'Auria-Fre+formulation+of+supergravity#Nieuwenhuizen82">can Nieuwenhuizen 1982</a>), which is a misnomer for what in mathematics are called <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a>s (since it is only the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/graded-commutative+algebra">graded-commutative algebra</a> that is required to be free, the <a class="existingWikiWord" href="/nlab/show/differential">differential</a> is crucially <em>not</em> free in general, otherwise one has just a <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>).</p> <p>The translation of <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fr%C3%A9+formulation+of+supergravity">D'Auria-Fré formalism</a> (“FDA”s) to explicit (<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra language was made in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="https://golem.ph.utexas.edu/category/2006/08/sugra_3connection_reloaded.html">SuGra 3-Connection Reloaded</a></em> (2006)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, example 5 in section 6.5.1, p. 54 of: <em><a class="existingWikiWord" href="/schreiber/show/L-infinity+algebra+connections"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math> algebra connections and applications to String- and Chern-Simons n-transport</a></em>, in: <em>Quantum Field Theory</em>, Birkhäuser (2009) 303-424 [<a href="http://arxiv.org/abs/0801.3480">arXiv:0801.3480</a>, <a href="https://doi.org/10.1007/978-3-7643-8736-5_17">doi:10.1007/978-3-7643-8736-5_17</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">Super Lie n-algebra extensions, higher WZW models and super p-branes with tensor multiplet fields</a></em>, Int. J. of Geometric Methods in Modern Physics <strong>12</strong> 02 (2015) 1550018 &lbrack;<a href="http://arxiv.org/abs/1308.5264">arXiv:1308.5264</a>, <a href="https://doi.org/10.1142/S0219887815500188">doi:10.1142/S0219887815500188</a>&rbrack;</p> </li> </ul> <p>connecting them to the <a class="existingWikiWord" href="/nlab/show/higher+WZW+terms">higher WZW terms</a> of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma+models">Green-Schwarz sigma models</a> of fundamental <a class="existingWikiWord" href="/nlab/show/super+p-branes">super p-branes</a> (<a class="existingWikiWord" href="/schreiber/show/The+brane+bouquet">The brane bouquet</a>).</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></em>, and <em><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></em>.</p> <p>Further exposition and review of the (dual) identification of supergravity “FDAs” with <a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="https://www.physicsforums.com/insights/homotopy-lie-n-algebras-supergravity/">Homotopy Lie n-algebras in Supergravity</a></em>, PhysicsForums-Insights (2015)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Saemann">Christian Saemann</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Wolf">Martin Wolf</a>: <em>Higher Structures in M-Theory</em>, Introduction to <em><a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a></em>, Fortsch. d. Phys. <strong>67</strong> 8-9 (2019) 1910001 &lbrack;<a href="https://arxiv.org/abs/1903.02807">arXiv:1903.02807</a>, <a href="https://doi.org/10.1002/prop.201910001">doi:10.1002/prop.201910001</a>&rbrack;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/The+rational+higher+structure+of+M-theory">The rational higher structure of M-theory</a></em>, in <em><a class="existingWikiWord" href="/nlab/show/Higher+Structures+in+M-Theory+2018">Higher Structures in M-Theory 2018</a></em>, Fortschr. der Physik <strong>67</strong> 8-9 (2019) 1910017 [<a href="https://arxiv.org/abs/1903.02834">arXiv:1903.02834</a>, <a href="https://doi.org/10.1002/prop.201910017">doi:10.1002/prop.201910017</a>]</p> </li> </ul> <p>Notice that there is a <em>different</em> concept of “Filipov <a class="existingWikiWord" href="/nlab/show/n-Lie+algebra">n-Lie algebra</a>” suggested by <a href="BLG+model#BaggerLambert06">Bagger& Lambert 2006</a> to play a role in the description of the <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> in the <a class="existingWikiWord" href="/nlab/show/near+horizon+limit">near horizon limit</a> of <a class="existingWikiWord" href="/nlab/show/black+p-branes">black p-branes</a>, notably the <a class="existingWikiWord" href="/nlab/show/BLG+model">BLG model</a> for the conformal <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> theory on the <a class="existingWikiWord" href="/nlab/show/M2-brane">M2-brane</a> .</p> <p>A realization of these “Filippov <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math>-Lie algebras” as 2-term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras (<a class="existingWikiWord" href="/nlab/show/Lie+2-algebras">Lie 2-algebras</a>) equipped with a binary <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> (“metric Lie 2-algebras”) is in:</p> <ul> <li> <p>Sam Palmer, <a class="existingWikiWord" href="/nlab/show/Christian+Saemann">Christian Saemann</a>, section 2 of <em>M-brane Models from Non-Abelian Gerbes</em>, JHEP 1207:010, 2012 (<a href="http://arxiv.org/abs/1203.5757">arXiv:1203.5757</a>)</p> </li> <li id="SaemannRitter13"> <p><a class="existingWikiWord" href="/nlab/show/Patricia+Ritter">Patricia Ritter</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Saemann">Christian Saemann</a>, section 2.5 of <em>Lie 2-algebra models</em>, JHEP 04 (2014) 066 (<a href="http://arxiv.org/abs/1308.4892">arXiv:1308.4892</a>)</p> </li> </ul> <p>based on</p> <ul> <li id="MFFMER08"><a class="existingWikiWord" href="/nlab/show/Paul+de+Medeiros">Paul de Medeiros</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, <a class="existingWikiWord" href="/nlab/show/Elena+M%C3%A9ndez-Escobar">Elena Méndez-Escobar</a>, <a class="existingWikiWord" href="/nlab/show/Patricia+Ritter">Patricia Ritter</a>, <em>On the Lie-algebraic origin of metric 3-algebras</em>, Commun.Math.Phys.290:871-902,2009 (<a href="http://arxiv.org/abs/0809.1086">arXiv:0809.1086</a>)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Figueroa-O%27Farrill">José Figueroa-O'Farrill</a>, section <em>Triple systems and Lie superalgebras</em> in <em>M2-branes, ADE and Lie superalgebras</em>, talk at IPMU 2009 (<a href="http://www.maths.ed.ac.uk/~jmf/CV/Seminars/Hongo.pdf">pdf</a>)</li> </ul> <div> <h4 id="ReferencesCFieldGaugeAlgebra">Supergravity C-Field gauge algebra</h4> <p>Identifying the super-graded gauge algebra of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">C-field</a> in <a class="existingWikiWord" href="/nlab/show/D%3D11+supergravity">D=11 supergravity</a> (with non-trivial <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>3</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>v</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">[v_3, v_3] = -v_6</annotation></semantics></math>):</p> <ul> <li id="CremmerJuliaLuPope"> <p><a class="existingWikiWord" href="/nlab/show/Eugene+Cremmer">Eugene Cremmer</a>, <a class="existingWikiWord" href="/nlab/show/Bernard+Julia">Bernard Julia</a>, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+Pope">Christopher Pope</a>, Equation (2.6) of <em>Dualisation of Dualities, II: Twisted self-duality of doubled fields and superdualities</em>, Nucl.Phys. B <strong>535</strong> (1998) 242-292 [<a href="https://doi.org/10.1016/S0550-3213(98)00552-5">doi:10.1016/S0550-3213(98)00552-5</a>, <a href="https://arxiv.org/abs/hep-th/9806106">arXiv:hep-th/9806106</a>]</p> </li> <li> <p>I. V. Lavrinenko, H. Lu, <a class="existingWikiWord" href="/nlab/show/Christopher+N.+Pope">Christopher N. Pope</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+S.+Stelle">Kellogg S. Stelle</a>, (3.4) in: <em>Superdualities, Brane Tensions and Massive IIA/IIB Duality</em>, Nucl. Phys. B <strong>555</strong> (1999) 201-227 [<a href="https://doi.org/10.1016/S0550-3213(99)00307-7">doi:10.1016/S0550-3213(99)00307-7</a>, <a href="https://arxiv.org/abs/hep-th/9903057">arXiv:hep-th/9903057</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jussi+Kalkkinen">Jussi Kalkkinen</a>, <a class="existingWikiWord" href="/nlab/show/Kellogg+S.+Stelle">Kellogg S. Stelle</a>, (75) of: <em>Large Gauge Transformations in M-theory</em>, J. Geom. Phys. <strong>48</strong> (2003) 100-132 [<a href="https://doi.org/10.1016/S0393-0440(03)00027-5">doi:10.1016/S0393-0440(03)00027-5</a>, <a href="https://arxiv.org/abs/hep-th/0212081">arXiv:hep-th/0212081</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+A.+Bandos">Igor A. Bandos</a>, <a class="existingWikiWord" href="/nlab/show/Alexei+J.+Nurmagambetov">Alexei J. Nurmagambetov</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+P.+Sorokin">Dmitri P. Sorokin</a>, (86) in: <em>Various Faces of Type IIA Supergravity</em>, Nucl.Phys. B <strong>676</strong> (2004) 189-228 [<a href="https://doi.org/10.1016/j.nuclphysb.2003.10.036">doi:10.1016/j.nuclphysb.2003.10.036</a>, <a href="https://arxiv.org/abs/hep-th/0307153">arXiv:hep-th/0307153</a>]</p> </li> </ul> <p>Identification as an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebra</a> (a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, in this case):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, (4.9) in: <em>Geometric and topological structures related to M-branes</em>, in <em>Superstrings, Geometry, Topology, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^\ast</annotation></semantics></math>-algebras</em>, Proc. Symp. Pure Math. <strong>81</strong> (2010) 181-236 [<a href="http://www.ams.org/books/pspum/081">ams:pspum/081</a>, <a href="http://arXiv.org/abs/1001.5020">arXiv:1001.5020</a>]</li> </ul> <p>and identificatoin with the rational <a class="existingWikiWord" href="/nlab/show/Whitehead+L-infinity+algebra">Whitehead <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebra</a> (the <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational Quillen model</a>) of the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> (cf. <em><a class="existingWikiWord" href="/schreiber/show/Hypothesis+H">Hypothesis H</a></em>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Voronov">Alexander Voronov</a>, (13) in: <em>Mysterious Triality and M-Theory</em> [<a href="https://arxiv.org/abs/2212.13968">arXiv:2212.13968</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, (22) in: <em><a class="existingWikiWord" href="/schreiber/show/Flux+Quantization+on+Phase+Space">Flux Quantization on Phase Space</a></em> [<a href="https://arxiv.org/abs/2312.12517">arXiv:2312.12517</a>]</p> </li> </ul> </div> <h4 id="ReferencesBVBRSTFormalism">In BV-BRST formalism</h4> <p>The introduction of <a class="existingWikiWord" href="/nlab/show/BV-BRST+complexes">BV-BRST complexes</a> as a model for the <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> of the <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> of <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> is due to</p> <ul> <li id="BatalinVilkovisky81"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Gauge Algebra and Quantization</em>, Phys. Lett. B 102 (1981) 27–31. doi:10.1016/0370-2693(81)90205-7</p> </li> <li id="BatalinVilkovisky83"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Feynman rules for reducible gauge theories</em>, Phys. Lett. B 120 (1983) 166-170.</p> <p>doi:10.1016/0370-2693(83)90645-7</p> </li> <li id="BatakinFradkin83"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Efim+Fradkin">Efim Fradkin</a>, <em>A generalized canonical formalism and quantization of reducible gauge theories</em>, Phys. Lett. B122 (1983) 157-164.</p> </li> <li id="BatalinVilkovisky83b"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Quantization of Gauge Theories with Linearly Dependent Generators</em>, Phys. Rev. D 28 (10): 2567–258 (1983) doi:10.1103/PhysRevD.28.2567. Erratum-ibid. 30 (1984) 508 doi:10.1103/PhysRevD.30.508</p> </li> </ul> <p>as reviewed in</p> <ul> <li id="HenneauxTeitelboim92"> <p><a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <a class="existingWikiWord" href="/nlab/show/Claudio+Teitelboim">Claudio Teitelboim</a>, <em><a class="existingWikiWord" href="/nlab/show/Quantization+of+Gauge+Systems">Quantization of Gauge Systems</a></em>, Princeton University Press 1992. xxviii+520 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joaquim+Gomis">Joaquim Gomis</a>, J. Paris, S. Samuel, <em>Antibrackets, Antifields and Gauge Theory Quantization</em> (<a href="http://arxiv.org/abs/hep-th/9412228">arXiv:hep-th/9412228</a>)</p> </li> </ul> <p>The understanding that these <a class="existingWikiWord" href="/nlab/show/BV-BRST+complexes">BV-BRST complexes</a> mathematically are the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of a <a class="existingWikiWord" href="/nlab/show/derived+L-%E2%88%9E+algebroid">derived L-∞ algebroid</a> originates around</p> <ul> <li id="Stasheff96"> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Homological Reduction of Constrained Poisson Algebras</em>, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 (<a href="http://arxiv.org/abs/q-alg/9603021">arXiv:q-alg/9603021</a>, <a href="https://projecteuclid.org/euclid.jdg/1214459757">Euclid</a>)</p> </li> <li id="Stasheff97"> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>The (secret?) homological algebra of the Batalin-Vilkovisky approach</em> (<a href="http://arxiv.org/abs/hep-th/9712157">arXiv:hep-th/9712157</a>)</p> </li> </ul> <p>Discussion in terms of homotopy <a class="existingWikiWord" href="/nlab/show/Lie-Rinehart+pairs">Lie-Rinehart pairs</a> is due to</p> <ul> <li id="Kjeseth01"><a class="existingWikiWord" href="/nlab/show/Lars+Kjeseth">Lars Kjeseth</a>, <em>Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs</em>, Homology Homotopy Appl. Volume 3, Number 1 (2001), 139-163. (<a href="https://projecteuclid.org/euclid.hha/1140370269">Euclid</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid">L-∞ algebroid</a>-structure is also made explicit in (<a href="http://arxiv.org/abs/0910.4001v1">def. 4.1 of v1</a>) of (<a href="#SatiSchreiberStasheff09">Sati-Schreiber-Stasheff 09</a>).</p> <p>The extraction of <a class="existingWikiWord" href="/nlab/show/L-infinity+algebras"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a> from the formal neighbourhood of a <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> is maybe first made explicit in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Grigoriev">Maxim Grigoriev</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Rudinsky">Dmitry Rudinsky</a>, <em>Notes on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-approach to local gauge field theories</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2303.08990">arXiv2303.08990</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <h4 id="in_string_field_theory">In string field theory</h4> <p>The first <em>explicit</em> appearance of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras in theoretical physics is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra structure on the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> found in the context of closed bosonic <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> in</p> <ul> <li id="Zwiebach92"> <p><a class="existingWikiWord" href="/nlab/show/Barton+Zwiebach">Barton Zwiebach</a>, <em>Closed string field theory: Quantum action and the B-V master equation</em> , Nucl.Phys. B390 (1993) 33 (<a href="http://arxiv.org/abs/hep-th/9206084">arXiv:hep-th/9206084</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space</em> Talk given at the <em>Conference on Topics in Geometry and Physics</em> (1992) (<a href="http://arxiv.org/abs/hep-th/9304061">arXiv:hep-th/9304061</a>)</p> </li> </ul> <p>Generalization to open-closed bosonic string field theory yields <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> interacting with <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hiroshige+Kajiura">Hiroshige Kajiura</a>, <em>Homotopy Algebra Morphism and Geometry of Classical String Field Theory</em> (2001) (<a href="http://arxiv.org/abs/hep-th/0112228">arXiv:hep-th/0112228</a>)</p> </li> <li id="KajiuraStasheff04"> <p><a class="existingWikiWord" href="/nlab/show/Hiroshige+Kajiura">Hiroshige Kajiura</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Homotopy algebras inspired by classical open-closed string field theory</em>, Comm. Math. Phys. 263 (2006) 553–581 (2004) (<a href="http://arxiv.org/abs/math/0410291">arXiv:math/0410291</a>)</p> </li> <li id="Markl"> <p><a class="existingWikiWord" href="/nlab/show/Martin+Markl">Martin Markl</a>, <em>Loop Homotopy Algebras in Closed String Field Theory</em> (1997) (<a href="http://arxiv.org/abs/hep-th/9711045">arXiv:hep-th/9711045</a>)</p> </li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Higher homotopy algebras: String field theory and Drinfeld’s quasiHopf algebras</em>, proceedings of <em>International Conference on Differential Geometric Methods in Theoretical Physics</em>, 1991 (<a href="https://inspirehep.net/record/327712">spire</a>)</li> </ul> <p>For more see at <em><a href="string%20field%20theory#ReferencesHomotopyAlgebra">string field theory – References – Relation to A-infinity and L-infinity algebras</a></em>.</p> <h4 id="in_deformation_quantization">In deformation quantization</h4> <p>The general solution of the <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> problem of <a class="existingWikiWord" href="/nlab/show/Poisson+manifolds">Poisson manifolds</a> due to</p> <ul> <li><a href="#Kontsevich97">Kontsevich 97</a></li> </ul> <p>makes crucial use of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>. Later it was understood that indeed <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> are equivalently the universal model for infinitesimal <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a> (of anything), also called <a class="existingWikiWord" href="/nlab/show/formal+moduli+problems">formal moduli problems</a>:</p> <ul> <li id="Hinich98"> <p><a class="existingWikiWord" href="/nlab/show/Vladimir+Hinich">Vladimir Hinich</a>, <em>DG coalgebras as formal stacks</em> (<a href="http://arxiv.org/abs/math/9812034">arXiv:9812034</a>)</p> </li> <li id="Lurie"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Formal+Moduli+Problems">Formal Moduli Problems</a></em></p> </li> <li id="Pridham"> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Pridham">Jonathan Pridham</a>, <em>Unifying derived deformation theories</em>, Adv. Math. 224 (2010), no.3, 772-826 (<a href="http://arxiv.org/abs/0705.0344">arXiv:0705.0344</a>)</p> </li> </ul> <h4 id="in_heterotic_string_theory">In heterotic string theory</h4> <p>Next it was again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> that drew attention. It was eventually understood that the <a class="existingWikiWord" href="/nlab/show/string+structures">string structures</a> which embody a refinement of the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+anomaly+cancellation">Green-Schwarz anomaly cancellation</a> mechanism in <a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a> have a further smooth refinement as <a class="existingWikiWord" href="/nlab/show/G-structures">G-structures</a> for the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, which is the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> called the <em><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></em>. This is due to</p> <ul> <li id="BCSS"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Alissa+Crans">Alissa Crans</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em>From loop groups to 2-groups</em>, <em>Homotopy, Homology and Applications</em> <strong>9</strong> (2007), 101-135. (<a href="http://arxiv.org/abs/math.QA/0504123">arXiv:math.QA/0504123</a>)</p> </li> <li id="Henriques"> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <em>Integrating <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math> algebras</em>, Compos. Math. <strong>144</strong> (2008), no. 4, 1017–1045 (<a href="http://dx.doi.org/10.1112/S0010437X07003405">doi</a>,<a href="http://arxiv.org/abs/math.AT/0603563">math.AT/0603563</a>)</p> </li> </ul> <p>and the relation to the Green-Schwarz mechanism is made explicit in</p> <ul> <li id="SatiSchreiberStasheff09"><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em>, Communications in Mathematical Physics, 2012, Volume 315, Issue 1, pp 169-213 (<a href="http://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>)</li> </ul> <p>This article also observes that an analogous situation appears in <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a> with the <em><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></em> in place of the string Lie 2-algebra.</p> <h4 id="higher_chernsimons_field_theory_and_aksz_sigmamodels">Higher Chern-Simons field theory and AKSZ sigma-models</h4> <p>Ordinary <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> for a simple gauge group is all controled by a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra 3-cocycle</a>. The generalization of Chern-Simons theory to <a class="existingWikiWord" href="/nlab/show/AKSZ-sigma+models">AKSZ-sigma models</a> was understood to be encoded by <a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroids">symplectic Lie n-algebroids</a> (later re-popularized as “<a class="existingWikiWord" href="/nlab/show/shifted+symplectic+structures">shifted symplectic structures</a>”) in</p> <ul> <li> <p><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>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavol+%C5%A0evera">Pavol Ševera</a>, <em><a class="existingWikiWord" href="/nlab/show/Some+title+containing+the+words+%22homotopy%22+and+%22symplectic%22%2C+e.g.+this+one">Some title containing the words "homotopy" and "symplectic", e.g. this one</a></em>, based on a talk at “Poisson 2000”, CIRM Marseille, June 2000; (<a href="http://arxiv.org/abs/math/0105080">arXiv:0105080</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dmitry+Roytenberg">Dmitry Roytenberg</a>, <em>On the structure of graded symplectic supermanifolds and Courant algebroids</em> in <em>Quantization, Poisson Brackets and Beyond</em> , <a class="existingWikiWord" href="/nlab/show/Theodore+Voronov">Theodore Voronov</a> (ed.), Contemp. Math., Vol. 315, Amer. Math. Soc., Providence, RI, 2002 (<a href="http://arxiv.org/abs/math/0203110">arXiv</a>)</p> </li> <li id="Roytenberg"> <p><a class="existingWikiWord" href="/nlab/show/Dmitry+Roytenberg">Dmitry Roytenberg</a>, <em>AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories</em> Lett.Math.Phys.79:143-159,2007 (<a href="http://arxiv.org/abs/hep-th/0608150">arXiv:hep-th/0608150</a>).</p> </li> </ul> <p>The globally defined AKSZ action functionals obtained this way were shown 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>to be a special case of the higher Lie integration process of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Cech+cocycles+for+differential+characteristic+classes">Cech cocycles for differential characteristic classes</a></em>, Advances in Theoretical and Mathematical Physics, Volume 16 Issue 1 (2012), pages 149-250 (<a href="http://arxiv.org/abs/1011.4735">arXiv:1011.4735</a>)</li> </ul> <p>Further exmaples of non-symplectic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-Chern-Simons theory obtained this way include <a class="existingWikiWord" href="/nlab/show/7-dimensional+Chern-Simons+theory">7-dimensional Chern-Simons theory</a> on <a class="existingWikiWord" href="/nlab/show/differential+string+structure">string 2-connections</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/7d+Chern-Simons+theory+and+the+5-brane">Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory</a></em>, Advances in Theoretical and Mathematical Physics, Volume 18, Number 2 (2014) p. 229–321</li> </ul> <h4 id="in_local_prequantum_field_theory">In local prequantum field theory</h4> <p>Infinite-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras that behaved similar to <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> – <em><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebras">Poisson bracket Lie n-algebras</a></em> – were noticed</p> <ul> <li id="Rogers10"> <p><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math> algebras from multisymplectic geometry</em> , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (<a href="http://arxiv.org/abs/1005.2230">arXiv:1005.2230</a>, <a href="http://link.springer.com/article/10.1007%2Fs11005-011-0493-x">journal</a>).</p> </li> <li id="Rogers11"> <p><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>)</p> </li> </ul> <p>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/L-%E2%88%9E+algebras+of+local+observables+from+higher+prequantum+bundles">L-∞ algebras of local observables from higher prequantum bundles</a></em>, Homology, Homotopy and Applications, Volume 16 (2014) Number 2, p. 107 – 142 (<a href="http://arxiv.org/abs/1304.6292">arXiv:1304.6292</a>)</li> </ul> <p>these were shown to be the infinitesimal version of the symmetries of <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundles">prequantum n-bundles</a> as they appear in <a class="existingWikiWord" href="/nlab/show/local+prequantum+field+theory">local prequantum field theory</a>, in higher generalization of how the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> is the Lie algebra of the <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>.</p> <p>These also encode a homotopy refinement of the <a class="existingWikiWord" href="/nlab/show/Dickey+bracket">Dickey bracket</a> on <a class="existingWikiWord" href="/nlab/show/Noether+theorem">Noether</a> <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a> which for <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+sigma+models">Green-Schwarz sigma models</a> reduces to 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>-algebras of <a class="existingWikiWord" href="/nlab/show/BPS+charges">BPS charges</a> which refine <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a> such as the <a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Lie+n-algebras+of+BPS+charges">Lie n-algebras of BPS charges</a></em> (<a href="http://arxiv.org/abs/1507.08692">arXiv:1507.08692</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Lie+n-algebras+of+higher+Noether+currents">Lie n-algebras of higher Noether currents</a></em></p> </li> </ul> <p>This makes concrete the suggestion that there should be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra refinements of the <a class="existingWikiWord" href="/nlab/show/Dickey+bracket">Dickey bracket</a> of <a class="existingWikiWord" href="/nlab/show/conserved+currents">conserved currents</a> in <a class="existingWikiWord" href="/nlab/show/local+field+theory">local field theory</a> that was made in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, <a class="existingWikiWord" href="/nlab/show/Ronald+Fulp">Ronald Fulp</a>, <a class="existingWikiWord" href="/nlab/show/Tom+Lada">Tom Lada</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>The sh Lie structure of Poisson brackets in field theory</em> (<a href="http://arxiv.org/abs/hep-th/9702176">arXiv:hep-th/9702176</a>)</li> </ul> <p>Comprehesive survey and exposition of this situation is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Higher+Prequantum+Geometry">Higher Prequantum Geometry</a></em>, in <a class="existingWikiWord" href="/nlab/show/Gabriel+Catren">Gabriel Catren</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+Anel">Mathieu Anel</a> (eds.), <em><a href="https://ncatlab.org/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, 2016</li> </ul> <h4 id="in_perturbative_quantum_field_theory">In perturbative quantum field theory</h4> <p>Further identification of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> in the <a class="existingWikiWord" href="/nlab/show/Feynman+amplitudes">Feynman amplitudes</a>/<a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> of <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian</a> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>:</p> <ul> <li id="Froeb18"> <p><a class="existingWikiWord" href="/nlab/show/Markus+Fr%C3%B6b">Markus Fröb</a>, <em>Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories</em> <a href="https://dx.doi.org/10.1007/s00220-019-03558-6">Communications in Mathematical Physics</a> 2019 (online first) (<a href="https://arxiv.org/abs/1803.10235">arXiv:1803.10235</a>)</p> </li> <li id="Arvanitakis19"> <p><a class="existingWikiWord" href="/nlab/show/Alex+Arvanitakis">Alex Arvanitakis</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra of the S-matrix</em> (<a href="https://arxiv.org/abs/1903.05643">arXiv:1903.05643</a>)</p> </li> </ul> <h4 id="in_double_field_theory">In double field theory</h4> <p>In <a class="existingWikiWord" href="/nlab/show/double+field+theory">double field theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andreas+Deser">Andreas Deser</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Even symplectic supermanifolds and double field theory</em>, Communications in Mathematical Physics November 2015, Volume 339, Issue 3, pp 1003-1020 (<a href="http://arxiv.org/abs/1406.3601">arXiv:1406.3601</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Olaf+Hohm">Olaf Hohm</a>, <a class="existingWikiWord" href="/nlab/show/Barton+Zwiebach">Barton Zwiebach</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math> Algebras and Field Theory</em> (<a href="https://arxiv.org/abs/1701.08824">arXiv:1701.08824</a>)</p> </li> </ul> <h3 id="related_expositions">Related expositions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory+FAQ">string theory FAQ</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motives+in+physics">motives in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert%27s+sixth+problem">Hilbert's sixth problem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+theory+and+physics">model theory and physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+higher+differential+geometry">motivation for higher differential geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesion">motivation for cohesion</a></p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on September 27, 2024 at 08:24:38. 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