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class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#cohomology'>Cohomology</a></li> <li><a href='#invariant_polynomials_and_chernsimons_elements'>Invariant polynomials and Chern-Simons elements</a></li> <li><a href='#lie_algebra_valued_forms'>Lie algebra valued forms</a></li> </ul> <li><a href='#related_structures'>Related structures</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>Poincaré Lie algebra</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(\mathbb{R}^{d-1,1})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> of the <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a> of <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>: the <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a>. This happens to be the <a class="existingWikiWord" href="/nlab/show/semidirect+product">semidirect product</a> of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(d-1,1)</annotation></semantics></math> with the the abelian translation Lie algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, regarded as the <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner product space</a> whose underlying <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math> and equipped with the <a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a> given in the canonical <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo>≔</mo><mi>diag</mi><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>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta \coloneqq diag(-1,+1,+1, \cdots, +1) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(\mathbb{R}^{d-1,1})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a> of this inner product space. The <em>Poincaré Lie algebra</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(\mathbb{R}^{d-1,1})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> of this <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> (its <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>≔</mo><mi>Lie</mi><mo stretchy="false">(</mo><mi>Iso</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{iso}(\mathbb{R}^{d-1,1}) \coloneqq Lie(Iso(\mathbb{R}^{d-1,1})) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+group">Poincaré group</a> is the <a class="existingWikiWord" href="/nlab/show/semidirect+product+group">semidirect product group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Iso</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>⋊</mo><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Iso(\mathbb{R}^{d-1,1}) \simeq \mathbb{R}^{d-1,1} \rtimes O(d-1,1) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,1)</annotation></semantics></math> (the group of <a class="existingWikiWord" href="/nlab/show/linear+map">linear</a> <a class="existingWikiWord" href="/nlab/show/isometries">isometries</a> of <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>) with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math> regarded as the <a class="existingWikiWord" href="/nlab/show/translation+group">translation group</a> along itself, via the defining <a class="existingWikiWord" href="/nlab/show/action">action</a>.</p> <p>Accordingly, the Poincaré Lie algebra is the <a class="existingWikiWord" href="/nlab/show/semidirect+product+Lie+algebra">semidirect product Lie algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo>⋊</mo><msup><mi>𝔰𝔬</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathfrak{iso}(\mathbb{R}^{d-1,1}) \simeq \mathbb{R}^{d-1,1} \rtimes \mathfrak{so}^+(d-1,1) </annotation></semantics></math></div> <p>of the abelian Lie algebra on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math> with the (orthochronous) <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(d-1,1)</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>P</mi> <mi>a</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{P_a\}</annotation></semantics></math> the canonical <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>L</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>L</mi> <mrow><mi>b</mi><mi>a</mi></mrow></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{L_{a b} = - L_{b a}\}</annotation></semantics></math> the corresponding canonical basis of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(d-1,1)</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(\mathbb{R}^{d-1,1})</annotation></semantics></math> is given as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">[</mo><msub><mi>P</mi> <mi>a</mi></msub><mo>,</mo><msub><mi>P</mi> <mi>b</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>L</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>,</mo><msub><mi>L</mi> <mrow><mi>c</mi><mi>d</mi></mrow></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><msub><mi>η</mi> <mrow><mi>d</mi><mi>a</mi></mrow></msub><msub><mi>L</mi> <mrow><mi>b</mi><mi>c</mi></mrow></msub><mo>−</mo><msub><mi>η</mi> <mrow><mi>b</mi><mi>c</mi></mrow></msub><msub><mi>L</mi> <mrow><mi>a</mi><mi>d</mi></mrow></msub><mo>+</mo><msub><mi>η</mi> <mrow><mi>a</mi><mi>c</mi></mrow></msub><msub><mi>L</mi> <mrow><mi>b</mi><mi>d</mi></mrow></msub><mo>−</mo><msub><mi>η</mi> <mrow><mi>d</mi><mi>b</mi></mrow></msub><msub><mi>L</mi> <mrow><mi>a</mi><mi>c</mi></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>L</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>,</mo><msub><mi>P</mi> <mi>c</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>=</mo><msub><mi>η</mi> <mrow><mi>a</mi><mi>c</mi></mrow></msub><msub><mi>P</mi> <mi>b</mi></msub><mo>−</mo><msub><mi>η</mi> <mi>bc</mi></msub><msub><mi>P</mi> <mi>a</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} [P_a, P_b] & = 0 \\ [L_{a b}, L_{c d}] & = \eta_{d a} L_{b c} -\eta_{b c} L_{a d} +\eta_{a c} L_{b d} -\eta_{d b} L_{a c} \\ [L_{a b}, P_c] & = \eta_{a c} P_b -\eta_{bc} P_a \end{aligned} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> sees only the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, and does not distinguish betwee a Lie group and any of its discrete <a class="existingWikiWord" href="/nlab/show/covering+spaces">covering spaces</a>, we may equivalently consider the Lie algebra of the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mi>SO</mi> <mo>+</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(d-1,1) \to SO^+(d-1,1)</annotation></semantics></math> (the double cover of the <a class="existingWikiWord" href="/nlab/show/proper+orthochronous+Lorentz+group">proper orthochronous Lorentz group</a>) and its <a class="existingWikiWord" href="/nlab/show/action">action</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math>.</p> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></em>, the Lie algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spin(d-1,1)</annotation></semantics></math> is the Lie algebra spanned by the <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a> <a class="existingWikiWord" href="/nlab/show/bivectors">bivectors</a></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> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>↔</mo><msub><mi>Γ</mi> <mi>a</mi></msub><msub><mi>Γ</mi> <mi>b</mi></msub></mrow><annotation encoding="application/x-tex"> L_{a b} \leftrightarrow \Gamma_a \Gamma_b </annotation></semantics></math></div> <p>and its <a class="existingWikiWord" href="/nlab/show/action">action</a> on itself as well as on the vectors, identified with single Clifford generators</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>a</mi></msub><mo>↔</mo><msub><mi>Γ</mi> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex"> P_a \leftrightarrow \Gamma_a </annotation></semantics></math></div> <p>is given by forming <a class="existingWikiWord" href="/nlab/show/commutators">commutators</a> in the <a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>L</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>,</mo><msub><mi>P</mi> <mi>c</mi></msub><mo stretchy="false">]</mo><mo>↔</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mi>c</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [L_{a b}, P_c] \leftrightarrow \tfrac{1}{2}[\Gamma_{a b}, \Gamma_c ] </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>L</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>,</mo><msub><mi>L</mi> <mrow><mi>c</mi><mi>d</mi></mrow></msub><mo stretchy="false">]</mo><mo>↔</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo>,</mo><msub><mi>Γ</mi> <mrow><mi>c</mi><mi>d</mi></mrow></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [L_{a b}, L_{c d}] \leftrightarrow \tfrac{1}{2}[\Gamma_{a b}, \Gamma_{c d} ] \,. </annotation></semantics></math></div> <p>Via the Clifford relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>a</mi></msub><msub><mi>Γ</mi> <mi>b</mi></msub><mo>+</mo><msub><mi>Γ</mi> <mi>b</mi></msub><msub><mi>Γ</mi> <mi>a</mi></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \Gamma_a \Gamma_b + \Gamma_b \Gamma_a = -2 \eta_{a b} </annotation></semantics></math></div> <p>this yields the claim.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Dually, the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{iso}(\mathbb{R}^{d-1})</annotation></semantics></math> is generated from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d,1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mn>2</mn></msup><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\wedge^2 \mathbb{R}^{d,1}</annotation></semantics></math>. For <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> the standard basis of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1}</annotation></semantics></math> we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\omega^{a b}\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>e</mi> <mi>a</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e^a\}</annotation></semantics></math> for these generators. With <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\eta_{a b})</annotation></semantics></math> the components of the <a class="existingWikiWord" href="/nlab/show/Minkowski+metric">Minkowski metric</a> we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>≔</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>c</mi></mrow></msup><msub><mi>η</mi> <mrow><mi>c</mi><mi>b</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega^{a}{}_b \coloneqq \omega^{a c}\eta_{c b} \,. </annotation></semantics></math></div> <p>In terms of this the CE-differential that defines the Lie algebra structure is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>=</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>c</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mrow><mi>c</mi><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> d_{CE} \colon \omega^{a b} = \omega^a{}_c \wedge \omega^{c b} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>CE</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>e</mi> <mi>a</mi></msup><mo>↦</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>t</mi> <mi>b</mi></msup></mrow><annotation encoding="application/x-tex"> d_{CE} \colon e^a \mapsto \omega^{a}{}_b \wedge t^b </annotation></semantics></math></div></div> <h2 id="properties">Properties</h2> <h3 id="cohomology">Cohomology</h3> <p>We discuss some elements in the <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(d-1,1)</annotation></semantics></math>.</p> <p>The canonical degree-3 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(d-1,1)</annotation></semantics></math>-cocycle is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mi>b</mi></msup><msub><mrow></mrow> <mi>c</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mi>c</mi></msup><msub><mrow></mrow> <mi>a</mi></msub><mo>∈</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a \in CE(\mathfrak{iso}(d-1,1)) \,. </annotation></semantics></math></div> <p>The <em>volume cocycle</em> is the <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a></p> <div id="VolumeCocycle" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>vol</mi><mo>=</mo><msub><mi>ϵ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>d</mi></msub></mrow></msub><msup><mi>e</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>e</mi> <mrow><msub><mi>a</mi> <mi>d</mi></msub></mrow></msup><mo>∈</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> vol = \epsilon_{a_1 \cdots a_{d}} e^{a_1} \wedge \cdots \wedge e^{a_d} \in CE(\mathfrak{iso}(d-1,1)) \,. </annotation></semantics></math></div> <h3 id="invariant_polynomials_and_chernsimons_elements">Invariant polynomials and Chern-Simons elements</h3> <p>With the basis elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>e</mi> <mi>a</mi></msup><mo>,</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e^a, \omega^{a b})</annotation></semantics></math> as above, denote the shifted generators of the <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(\mathfrak{iso}(d-1,1))</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>θ</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\theta^a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup></mrow><annotation encoding="application/x-tex">r^{a b}</annotation></semantics></math>, respectively.</p> <p>We have the <a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>W</mi></msub><mo>:</mo><msup><mi>r</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>↦</mo><msup><mi>ω</mi> <mrow><mi>a</mi><mi>c</mi></mrow></msup><mo>∧</mo><msub><mi>R</mi> <mi>c</mi></msub><msup><mrow></mrow> <mi>d</mi></msup><mo>−</mo><msup><mi>R</mi> <mrow><mi>a</mi><mi>c</mi></mrow></msup><mo>∧</mo><msub><mi>ω</mi> <mi>c</mi></msub><msup><mrow></mrow> <mi>b</mi></msup></mrow><annotation encoding="application/x-tex"> d_W : r^{a b} \mapsto \omega^{a c} \wedge R_c{}^d - R^{a c} \wedge \omega_c{}^b </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>W</mi></msub><mo>:</mo><msup><mi>θ</mi> <mi>a</mi></msup><mo>↦</mo><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><msup><mi>θ</mi> <mi>b</mi></msup><mo>−</mo><msup><mi>R</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><msup><mi>e</mi> <mi>b</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_W : \theta^a \mapsto \omega^a{}_b \theta^b - R^{a}{}_b e^b \,. </annotation></semantics></math></div> <p>The element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><msup><mi>θ</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>θ</mi> <mi>b</mi></msup><mo>∈</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_{a b} \theta^a \wedge \theta^b \in W(\mathfrak{iso}(d-1,1))</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a>. A <a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a> for it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cs</mi><mo>=</mo><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub><msup><mi>e</mi> <mi>a</mi></msup><mo>∧</mo><msup><mi>θ</mi> <mi>b</mi></msup></mrow><annotation encoding="application/x-tex">cs = \eta_{a b} e^a \wedge \theta^b</annotation></semantics></math>. So this transgresses to the trivial cocycle.</p> <p>Another invariant polynomial is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>∧</mo><msub><mi>r</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">r^{a b} \wedge r_{a b}</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{so}(d-1,1)</annotation></semantics></math>. Accordingly, it transgresses to a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ω</mi> <mi>a</mi></msup><msub><mrow></mrow> <mi>b</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mi>b</mi></msup><msub><mrow></mrow> <mi>c</mi></msub><mo>∧</mo><msup><mi>ω</mi> <mi>c</mi></msup><msub><mrow></mrow> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\omega^a{}_b \wedge \omega^b{}_c \wedge \omega^c{}_a</annotation></semantics></math>.</p> <p>This is the first in an infinite series of Pontryagin invariant polynomials</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>n</mi></msub><mo>:</mo><mo>=</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>3</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P_n := r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_n}{}_{a_1} \,. </annotation></semantics></math></div> <p>There is also an infinite series of mixed invariant polynomials</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>:</mo><mo>=</mo><msub><mi>θ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>3</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow></msub><mo>∧</mo><msup><mi>θ</mi> <mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_{2n + 2} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge \theta^{a_n} \,. </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+element">Chern-Simons element</a>s for these are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>:</mo><mo>=</mo><msub><mi>θ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msub><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msub><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mn>2</mn></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mn>3</mn></msub></mrow></msub><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>r</mi> <mrow><msub><mi>a</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msup><msub><mrow></mrow> <mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow></msub><mo>∧</mo><msup><mi>e</mi> <mrow><msub><mi>a</mi> <mi>n</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> B_{2n + 1} := \theta_{a_1} \wedge r^{a_1}{}_{a_2} \wedge r^{a_2}{}_{a_3} \wedge \cdots \wedge r^{a_{n-1}}{}_{a_n} \wedge e^{a_n} \,. </annotation></semantics></math></div> <h3 id="lie_algebra_valued_forms">Lie algebra valued forms</h3> <p>A <a class="existingWikiWord" href="/nlab/show/Lie+algebra-valued+form">Lie algebra-valued form</a> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{iso}(d-1,1)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>←</mo><mi>W</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>Ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(X) \leftarrow W(\mathfrak{iso}(d-1,1)) : (E,\Omega) </annotation></semantics></math></div> <p>is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a “<a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a>” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>T</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(T, R)</annotation></semantics></math> consists of</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/torsion+of+a+metric+connection">torsion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mi>d</mi><mi>E</mi><mo>+</mo><mo stretchy="false">[</mo><mi>Ω</mi><mo>∧</mo><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">T = d E + [\Omega \wedge E]</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Riemannian+curvature">Riemannian curvature</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>d</mi><mi>Ω</mi><mo>+</mo><mo stretchy="false">[</mo><mi>Ω</mi><mo>∧</mo><mi>Ω</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">R = d \Omega + [\Omega \wedge \Omega]</annotation></semantics></math>.</p> </li> </ul> <p>If the torsion vanishes, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Levi-Civita+connection">Levi-Civita connection</a> for the <a class="existingWikiWord" href="/nlab/show/metric">metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>a</mi></msup><mo>⊗</mo><msup><mi>E</mi> <mi>b</mi></msup><msub><mi>η</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msub></mrow><annotation encoding="application/x-tex">E^a \otimes E^b \eta_{a b}</annotation></semantics></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> is the image of the <a href="#VolumeCocycle">volume cocycle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>Ω</mi><mo stretchy="false">)</mo></mrow></mover><mi>W</mi><mo stretchy="false">(</mo><mi>𝔦𝔰𝔬</mi><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>←</mo><mi>vol</mi></mover><mi>W</mi><mo stretchy="false">(</mo><msup><mi>b</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>vol</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(X) \stackrel{(E,\Omega)}{\leftarrow} W(\mathfrak{iso}(d-1,1)) \stackrel{vol}{\leftarrow} W(b^{d-1} \mathbb{R}) : vol(E) \,. </annotation></semantics></math></div> <p>We have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>vol</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ϵ</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>a</mi> <mi>d</mi></msub></mrow></msub><msup><mi>E</mi> <mrow><msub><mi>a</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>E</mi> <mrow><msub><mi>a</mi> <mi>d</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> vol(E) = \epsilon_{a_1 \cdots a_d} E^{a_1} \wedge \cdots \wedge E^{a_d} \,. </annotation></semantics></math></div> <p>If the torsion vanishes, this is indeed a closed form.</p> <h2 id="related_structures">Related structures</h2> <ul> <li> <p><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/Poincar%C3%A9-Weyl+algebra">Poincaré-Weyl algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+Lie+algebra">super Poincaré Lie algebra</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 2, 2024 at 18:13:04. 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