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<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> <ul> <li><a href='#ordinary_definition'>Ordinary definition</a></li> <li><a href='#InAGeneralLinearCategory'>Internal to a general linear category</a></li> <li><a href='#GeneralAbstractPerspective'>General abstract perspective</a></li> </ul> <li><a href='#extra_stuff_structure_properties'>Extra stuff, structure, properties</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#cohomology'>Cohomology</a></li> <li><a href='#lie_theory_2'>Lie theory</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Lie algebra</em> is the <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a> approximation to a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>.</p> <h2 id="definition">Definition</h2> <h3 id="ordinary_definition">Ordinary definition</h3> <p>A <em>Lie algebra</em> is a <a class="existingWikiWord" href="/nlab/show/vector+space">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 a bilinear skew-symmetric map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>:</mo><mi>𝔤</mi><mo>∧</mo><mi>𝔤</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">[-,-] : \mathfrak{g} \wedge \mathfrak{g} \to \mathfrak{g}</annotation></semantics></math> which satisfies the <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∀</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>𝔤</mi><mo>:</mo><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mrow><mo>[</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mi>z</mi><mo>,</mo><mrow><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mi>y</mi><mo>,</mo><mrow><mo>[</mo><mi>z</mi><mo>,</mo><mi>x</mi><mo>]</mo></mrow><mo>]</mo></mrow><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \forall x,y,z \in \mathfrak{g} : \left[x,\left[y,z\right]\right] + \left[z,\left[x,y\right]\right] + \left[y,\left[z,x\right]\right] = 0 \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of Lie algebras is a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>𝔤</mi><mo>→</mo><mi>𝔥</mi></mrow><annotation encoding="application/x-tex">\phi : \mathfrak{g} \to \mathfrak{h}</annotation></semantics></math> such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">x,y \in \mathfrak{g}</annotation></semantics></math> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><msub><mo stretchy="false">]</mo> <mi>𝔤</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">]</mo> <mi>𝔥</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi([x,y]_{\mathfrak{g}}) = [\phi(x),\phi(y)]_{\mathfrak{h}} \,. </annotation></semantics></math></div> <p>This defines the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/LieAlg">LieAlg</a> of Lie algebras.</p> <h3 id="InAGeneralLinearCategory">Internal to a general linear category</h3> <p>The notion of <em><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></em> may be formulated <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> any <a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a>. This general definition subsumes variants of Lie algebras such as <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a>.</p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/commutative+unital+ring">commutative unital ring</a> <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>, and a (strict for simplicity) <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <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/linear+category">linear category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes,1)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>.</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie algebra object</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes,1,\tau)</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> L \in \mathcal{C} </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> (the <a class="existingWikiWord" href="/nlab/show/Lie+bracket">Lie bracket</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>L</mi><mo>⊗</mo><mi>L</mi><mo>→</mo><mi>L</mi></mrow><annotation encoding="application/x-tex"> [-,-] \;\colon\; L \otimes L \to L </annotation></semantics></math></div></li> </ol> <p>such that the following conditions hold:</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow><mo>]</mo></mrow><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>L</mi></msub><mo>⊗</mo><msub><mi>τ</mi> <mrow><mi>L</mi><mo>,</mo><mi>L</mi></mrow></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>τ</mi><mo>⊗</mo><msub><mi>id</mi> <mi>L</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mrow><mo>[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>]</mo></mrow><mo>]</mo></mrow><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>τ</mi> <mrow><mi>L</mi><mo>,</mo><mi>L</mi></mrow></msub><mo>⊗</mo><msub><mi>id</mi> <mi>L</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>L</mi></msub><mo>⊗</mo><msub><mi>τ</mi> <mrow><mi>L</mi><mo>,</mo><mi>L</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> \left[-,\left[-,-\right]\right] + \left[-,\left[-,-\right]\right] \circ(id_L\otimes\tau_{L,L}) \circ(\tau\otimes id_L) + \left[-,\left[-,-\right]\right] \circ (\tau_{L,L}\otimes id_L)\circ (id_L\otimes\tau_{L,L}) = 0 </annotation></semantics></math></div></li> <li> <p>skew-symmetry:</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></mtd> <mtd><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo lspace="verythinmathspace" rspace="0em">+</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><msub><mi>τ</mi> <mrow><mi>L</mi><mo>,</mo><mi>L</mi></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><mn>0</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & \phantom{+} [-,-] \\ & + [-,-]\circ \tau_{L,L} \\ & = \phantom{+} 0 \end{aligned} </annotation></semantics></math></div></li> </ol> <p>Equivalently, Lie algebra objects are the <a class="existingWikiWord" href="/nlab/show/algebras+over+an+operad">algebras over an operad</a> over a certain quadratic <a class="existingWikiWord" href="/nlab/show/operad">operad</a>, called the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a>, which is the <a class="existingWikiWord" href="/nlab/show/Koszul+dual">Koszul dual</a> of the <a class="existingWikiWord" href="/nlab/show/commutative+algebra+operad">commutative algebra operad</a>.</p> <p>Examples of types of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+objects">Lie algebra objects</a>:</p> <p>If <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> is the ring <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> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = </annotation></semantics></math> <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/Mod">Mod</a> = <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> is the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a>, then a Lie algebra object is called a <em><a class="existingWikiWord" href="/nlab/show/Lie+ring">Lie ring</a></em>.</p> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> is the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> 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> equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> then a Lie algebra object is an ordinary_<a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie k-algebra</a><em>.</em></p> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/field">field</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> = <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a> is the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> 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>, then a Lie algebra object is a <em><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></em>.</p> <h3 id="GeneralAbstractPerspective">General abstract perspective</h3> <p>Lie algebras are equivalently groups in “<a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> geometry”.</p> <p>For instance in <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> then a Lie algebra of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> is just the first-order infinitesimal neighbourhood of the unit element (e.g. <a href="#Kock09">Kock 09, section 6</a>).</p> <p>More generally in geometric <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, Lie algebras, being 0-truncated <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> are equivalently “infinitesimal <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a>” (e.g. <a href="infinitesimal+cohesive+(infinity,1">here</a>-topos#FormalModuliProblems)), also called <a class="existingWikiWord" href="/nlab/show/formal+moduli+problems">formal moduli problems</a> (see there for more).</p> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <p>Notions of Lie algebras with extra <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">stuff, structure, property</a> includes</p> <ul> <li> <p>extra property</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+Lie+algebra">abelian Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simple+Lie+algebra">simple Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reductive+Lie+algebra">reductive Lie algebra</a></p> </li> </ul> </li> <li> <p>extra structure</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> <li> <p>extra stuff</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/PBW+theorem">PBW theorem</a></li> </ul> <h3 id="cohomology">Cohomology</h3> <p>See <a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>.</p> <h3 id="lie_theory_2">Lie theory</h3> <p>See</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></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> </ul> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/translation+Lie+algebra">translation Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/line+Lie+algebra">line Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivation+Lie+algebra">derivation Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+Lie+algebra">endomorphism Lie algebra</a></p> </li> <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></p> </li> <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/special+unitary+Lie+algebra">special unitary Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+Lie+algebra">Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><strong>Lie algebra</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+Lie+algebra">abelian</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple</a>, <a class="existingWikiWord" href="/nlab/show/reductive+Lie+algebra">reductive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+Lie+algebra">metric Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+enveloping+algebra">universal enveloping algebra</a>, <a class="existingWikiWord" href="/nlab/show/Cartan+subalgebra">Cartan subalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/root+%28in+representation+theory%29">root (in representation theory)</a>, <a class="existingWikiWord" href="/nlab/show/weight+%28in+representation+theory%29">weight (in representation theory)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+representation">Lie algebra representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+ideal">Lie ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley+basis">Chevalley basis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+algebra">Kac-Moody algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+Lie+algebra">complex Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/restricted+Lie+algebra">restricted Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie-Poisson+structure">Lie-Poisson structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+extension">Lie algebra extension</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semidirect+product+Lie+algebra">semidirect product Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+contraction">Lie algebra contraction</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Leibniz+algebra">Leibniz algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebroid">L-∞ algebroid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+coalgebra">Lie coalgebra</a></p> </li> </ul> <div> <p><strong>Examples of sequences of local structures</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>point</th><th>first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> = arbitrary order infinitesimal</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>local = <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>finite</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <strong><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a> (path)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tangent+vector">tangent vector</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/jet">jet</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a> of <a class="existingWikiWord" href="/nlab/show/curve">curve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ+of+a+space">germ of a space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/square-0+ring+extension">square-0 ring extension</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nilpotent+ring+extension">nilpotent ring extension</a>/<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{(p)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization at (p)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/integers">integers</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></td><td style="text-align: left;"></td><td style="text-align: left;">local strict deformation quantization</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Monographs:</p> <ul> <li id="Serre64"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>: <em>Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University</em>, Lecture Notes in Mathematics <strong>1500</strong>, Springer (1992) [<a href="https://doi.org/10.1007/978-3-540-70634-2">doi:10.1007/978-3-540-70634-2</a>]</p> </li> <li> <p>Arthur A. Sagle, Ralph E. Walde: <em>Introduction to Lie Groups and Lie Algebras</em>, Pure and Applied Mathematics <strong>51</strong>, Elsevier (1973) 215-227</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, <em>Lie groups and Lie algebras – Chapters 1-3</em>, Springer (1975, 1989) [<a href="https://link.springer.com/book/9783540642428">ISBN:9783540642428</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerhard+P.+Hochschild">Gerhard P. Hochschild</a>, <em>Basic Theory of Algebraic Groups and Lie Algebras</em>, Graduate Texts in Mathematics <strong>75</strong>, Springer (1981) [<a href="https://doi.org/10.1007/978-1-4613-8114-3_16">doi:10.1007/978-1-4613-8114-3_16</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, Ch. I of: <em>Representations of compact Lie groups</em>, Springer (1985) [<a href="https://link.springer.com/book/10.1007/978-3-662-12918-0">doi:10.1007/978-3-662-12918-0</a>]</p> <blockquote> <p>(in the context of <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M.+M.+Postnikov">M. M. Postnikov</a>, <em>Lectures on geometry: Semester V, Lie groups and algebras</em> (1986) [<a href="https://archive.org/details/postnikov-lectures-in-geometry-semester-v-lie-group-and-lie-algebras">ark:/13960/t4cp9jn4p</a>]</p> </li> <li id="Onishchik93"> <p>A. L. Onishchik (ed.) <em>Lie Groups and Lie Algebras</em></p> <ul> <li> <p><em>I.</em> A. L. Onishchik, E. B. Vinberg, <em>Foundations of Lie Theory</em>,</p> </li> <li> <p><em>II.</em> V. V. Gorbatsevich, A. L. Onishchik, <em>Lie Transformation Groups</em></p> </li> </ul> <p>Encyclopaedia of Mathematical Sciences <strong>20</strong>, Springer (1993)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+de+Azc%C3%A1rraga">José de Azcárraga</a>, <a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+M.+Izquierdo">José M. Izquierdo</a>, <em><a class="existingWikiWord" href="/nlab/show/Lie+Groups%2C+Lie+Algebras%2C+Cohomology+and+Some+Applications+in+Physics">Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics</a></em>, Cambridge Monographs of Mathematical Physics, Cambridge University Press (1995) [<a href="https://doi.org/10.1017/CBO9780511599897">doi:10.1017/CBO9780511599897</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Howard+Georgi">Howard Georgi</a>, <em>Lie Algebras In Particle Physics</em>, Westview Press (1999), CRC Press (2019) [<a href="https://doi.org/10.1201/9780429499210">doi:10.1201/9780429499210</a>]</p> <blockquote> <p>with an eye towards application to (the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model</a> of) <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hans+Duistermaat">Hans Duistermaat</a>, <a class="existingWikiWord" href="/nlab/show/Johan+A.+C.+Kolk">Johan A. C. Kolk</a>, Chapter 1 of: <em>Lie groups</em>, Springer (2000) [<a href="https://doi.org/10.1007/978-3-642-56936-4">doi:10.1007/978-3-642-56936-4</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Shlomo+Sternberg">Shlomo Sternberg</a>: <em>Lie Algebras</em> (2004) [<a href="https://people.math.harvard.edu/~shlomo/docs/lie_algebras.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Sternberg-LieAlgebras.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckhard+Meinrenken">Eckhard Meinrenken</a>, <em>Lie groups and Lie algebas</em>, Lecture notes (2010) [<a href="http://www.math.toronto.edu/mein/teaching/lie.pdf">pdf</a>]</p> </li> <li id="Hall15"> <p><a class="existingWikiWord" href="/nlab/show/Brian+C.+Hall">Brian C. Hall</a>, <em>Lie Groups, Lie Algebras, and Representations</em>, Springer 2015 (<a href="https://doi.org/10.1007/978-3-319-13467-3">doi:10.1007/978-3-319-13467-3</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Woit">Peter Woit</a>, Ch. 5 of <em>Quantum Theory, Groups and Representations: An Introduction</em>, Springer 2017 [<a href="https://doi.org/10.1007/978-3-319-64612-1">doi:10.1007/978-3-319-64612-1</a>, ISBN:978-3-319-64610-7]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <em>Lie groups and Lie algebras</em> [<a href="https://arxiv.org/abs/2201.09397">arXiv:2201.09397</a>]</p> </li> </ul> <p>Discussion with a view towards <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Werner+Greub">Werner Greub</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Ray+Vanstone">Ray Vanstone</a>, chapter IV in vol III of: <em><a class="existingWikiWord" href="/nlab/show/Connections%2C+Curvature%2C+and+Cohomology">Connections, Curvature, and Cohomology</a></em> Academic Press (1973)</li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> is in</p> <ul> <li id="Kock09"><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, section 6 of <em>Synthetic Geometry of Manifolds</em>, 2009 (<a href="http://home.imf.au.dk/kock/SGM-final.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 3, 2024 at 10:13:05. 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