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class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </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>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <ul> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> </ul> <li><a href='#refinements_and_generalizations'>Refinements and generalizations</a></li> <ul> <li><a href='#CartanGeometry'>From local to global geometry – Cartan geometry</a></li> <li><a href='#higher_klein_geometry'>Higher Klein geometry</a></li> <li><a href='#logicality_and_invariance'>Logicality and invariance</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#kleins_original_and_historical_context'>Klein’s original and historical context</a></li> <li><a href='#related_commentary'>Related commentary</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>The <em>Erlangen program</em> (in German, <em>Erlanger Programm</em> ) is a project, begun by <a class="existingWikiWord" href="/nlab/show/Felix+Klein">Felix Klein</a> at Erlangen in the 19th century (<a href="#Klein1872">Klein 1872</a>), to study <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> via <a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a> <a class="existingWikiWord" href="/nlab/show/groups">groups</a> of “geometric shapes”, hence from the perspective of <a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a>. The idea is to take the elementary building blocks of geometry to be not just <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a> but more generally <a class="existingWikiWord" href="/nlab/show/homogeneous+spaces">homogeneous spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>. These have then also been called <em><a class="existingWikiWord" href="/nlab/show/Klein+geometries">Klein geometries</a></em> .</p> <p>In (<a href="#Klein1872">Klein 1872, section 1</a>) the theme of the program is described as follows:</p> <blockquote> <p>Given a manifold and a transformation group acting on it, to investigate those properties of figures [Gebilde] on that manifold which are invariant under [all] transformations of that group.</p> </blockquote> <p>In modern language this means to consider a group of <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a> (<a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a>) <a class="existingWikiWord" href="/nlab/show/action">acting</a> on a (<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a>) <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> together with its <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> of any prescribed <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a>. (The concept of <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> only emerged at that time, in fact Klein and <a class="existingWikiWord" href="/nlab/show/Sophus+Lie">Sophus Lie</a> were in close contact, see <a href="#BirkhoffBennett">Birkhoff-Bennett</a>.)</p> <p>The group of all such transformations</p> <blockquote> <p>by which the geometric properties of configurations in space remain entirely unchanged</p> </blockquote> <p>is called the “Hauptgruppe”, translated to “principal group”.</p> <p>A few lines below in (<a href="#Klein1872">Klein 1872, section 1</a>) is the converse statement</p> <blockquote> <p>Given a manifold, and a transformation group acting on it, to study its invariants.</p> </blockquote> <p>Hence to find the figures which are left invariant by a given group <a class="existingWikiWord" href="/nlab/show/action">action</a>.</p> <p>Then in (<a href="#Klein1872">Klein 1872, end of section 5</a>) it says:</p> <blockquote> <p>Suppose in space some group or other, the principal group for instance, be given. Let us then select a single configuration, say a point, or a straight line, or even an ellipsoid, etc., and apply to it all the transformations of the principal group. We thus obtain an infinite manifoldness with a number of dimensions in general equal to the number of arbitrary parameters contained in the group, but reducing in special cases, namely, when the configuration originally selected has the property of being transformed into itself by an infinite number of the transformations of the group. Every manifoldness generated in this way may be called, with reference to the generating group, a body.</p> </blockquote> <p>This means in modern language, that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the given group acting on a given space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S \hookrightarrow X</annotation></semantics></math> is a given subspace (a configuration), then the “body” (“Körper”) generated by this is the <a class="existingWikiWord" href="/nlab/show/coset">coset</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mi>Stab</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G/Stab_G(S) </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> of that configuration. In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a point we would now call this the <a class="existingWikiWord" href="/nlab/show/orbit">orbit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. See also there at <em><a href="stabilizer%20group#KleinGeometry">Stabilizer of shapes – Klein geometry</a></em>.</p> <p>The text goes on to argue that spaces of this form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mi>Stab</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G/Stab_G(S)</annotation></semantics></math> are of fundamental importance:</p> <blockquote> <p>If now we desire to base our investigations upon the group, selecting at the same time certain definite configurations as space-elements, and if we wish to represent uniformly things which are of like characteristics, we must evidently choose our space-elements in such a way that their manifoldness either is itself a body or can be decomposed into bodies.</p> </blockquote> <p>Following this, such <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> have come to also be called <em><a class="existingWikiWord" href="/nlab/show/Klein+geometries">Klein geometries</a></em>.</p> <p>When it was proposed, the Erlangen program served to unify various different kinds of geometry, discovered and studied at that time, into a common framwork. On the other hand, many kinds of geometries without global symmetries are not <a class="existingWikiWord" href="/nlab/show/Klein+geometries">Klein geometries</a>, notably <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a> is (in general) not. But a (<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a>) <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> is <em>locally</em> (tangentially) modeled on <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> (<a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>) and this local model space <em>is</em> a Klein geometry. The generalization of Klein geometry to such local situations is <em><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></em>, see <a href="#CartanGeometry">below</a>.</p> <h3 id="in_homotopy_type_theory">In homotopy type theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> the idea of a group of symmetries preserving a figure inside the larger group of symmetries acting on what the figure is inscribed in is represented by any map of the form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>H</mi><mo>→</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex"> B H \to B G </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of such a map is the Klein space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math>.</p> <h2 id="refinements_and_generalizations">Refinements and generalizations</h2> <h3 id="CartanGeometry">From local to global geometry – Cartan geometry</h3> <p>While many types of geometries (such as <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a>) are not in general <a class="existingWikiWord" href="/nlab/show/Klein+geometries">Klein geometries</a>, they are <em>locally</em> like Klein geometries. This generalization of Klein geometry is known as <em><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></em>.</p> <table><thead><tr><th>local model</th><th>global geometry</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+2-geometry">Klein 2-geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+2-geometry">Cartan 2-geometry</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></td></tr> </tbody></table> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> terminology this corresponds to “locally gauging” the <a class="existingWikiWord" href="/nlab/show/symmetry+group">symmetry group</a>. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math> the inclusion of the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> into the <a class="existingWikiWord" href="/nlab/show/Poincare+group">Poincare group</a>, then the corresponding <a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a> is just <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, but a corresponding <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> is any <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo-Riemannian manifold</a>, i.e. a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> characterized in the <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a>.</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> context</th><th><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a></th><th><a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a></th><th>local model <a class="existingWikiWord" href="/nlab/show/space">space</a></th><th>local <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>global <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/first+order+formulation+of+gravity">first order formulation of gravity</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>/<a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</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">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> (<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (“<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+connection">Cartan connection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+group">Euclidean 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><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(d)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rotation+group">rotation 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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euclidean+gravity">Euclidean gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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><mi>d</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Iso(d-1,1)</annotation></semantics></math></td><td style="text-align: left;"><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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lorentzian+geometry">Lorentzian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo-Riemannian geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Einstein+gravity">Einstein gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+group">anti de Sitter 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>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d-1,2)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti+de+Sitter+spacetime">anti de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>AdS</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">AdS^d</annotation></semantics></math></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/AdS+gravity">AdS gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Sitter+group">de Sitter 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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,1)</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Sitter+spacetime">de Sitter spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>dS</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">dS^d</annotation></semantics></math></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/deSitter+gravity">deSitter gravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+algebraic+group">linear algebraic group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/parabolic+subgroup">parabolic subgroup</a>/<a class="existingWikiWord" href="/nlab/show/Borel+subgroup">Borel subgroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/parabolic+geometry">parabolic geometry</a></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;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+group">conformal 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><mi>t</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(d,t+1)</annotation></semantics></math></td><td style="text-align: left;">conformal parabolic subgroup</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+space">Möbius space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>d</mi><mo>,</mo><mi>t</mi></mrow></msup></mrow><annotation encoding="application/x-tex">S^{d,t}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+connection">conformal connection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+gravity">conformal gravity</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</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">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> (<a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (“<a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Klein+geometry">super Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super Cartan geometry</a></td><td style="text-align: left;">Cartan <a class="existingWikiWord" href="/nlab/show/superconnection">superconnection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Poincar%C3%A9+group">super Poincaré group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a> <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><mo stretchy="false">|</mo><mi>N</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{d-1,1\vert N}</annotation></semantics></math></td><td style="text-align: left;">Lorentzian <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superconnection">superconnection</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+group">super anti de Sitter group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+anti+de+Sitter+spacetime">super anti de Sitter spacetime</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+2-group">smooth 2-group</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">G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-monomorphism">2-monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G//H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Klein+2-geometry">Klein 2-geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cartan+2-geometry">Cartan 2-geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-monomorphism">∞-monomorphism</a> (i.e. any <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \to G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G//H</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+Cartan+connection">higher Cartan connection</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">examples</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+superspacetime">extended super Minkowski spacetime</a></td><td style="text-align: left;"></td><td style="text-align: left;">extended supergeometry</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher</a> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>: <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II</a>, <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic</a>, <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11d</a></td></tr> </tbody></table> </div> <h3 id="higher_klein_geometry">Higher Klein geometry</h3> <p>Aspects of Klein geometry may be generalized from groups to <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> and even <a class="existingWikiWord" href="/nlab/show/category">categories</a> or <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>-<a class="existingWikiWord" href="/nlab/show/infinity-groupoid">groupoids</a>. See at <em><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></em>.</p> <h3 id="logicality_and_invariance">Logicality and invariance</h3> <p>Logicians have attempted to demonstrate that specifically <em>logical</em> constructions are those invariant under the largest group of transformations, in the sense of the Erlangen program. See <a class="existingWikiWord" href="/nlab/show/logicality+and+invariance">logicality and invariance</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a>, <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystallographic+group">crystallographic group</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="kleins_original_and_historical_context">Klein’s original and historical context</h3> <p>The original article:</p> <ul> <li id="Klein1872"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Klein">Felix Klein</a>, <em><a class="existingWikiWord" href="/nlab/show/Vergleichende+Betrachtungen+%C3%BCber+neuere+geometrische+Forschungen">Vergleichende Betrachtungen über neuere geometrische Forschungen</a></em> (1872) Mathematische Annalen volume 43, pages 63–100 1893 (<a href="https://doi.org/10.1007/BF01446615">doi:10.1007/BF01446615</a>)</p> <p>English translation by M. W. Haskell:</p> <p><em>A comparative review of recent researches in geometry</em>, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (<a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full">euclid:1183407629</a>, LaTeX version retyped by Nitin C. Rughoonauth: <a href="https://arxiv.org/abs/0807.3161">arXiv:0807.3161</a>)</p> </li> </ul> <p>Historical context and related material:</p> <ul> <li id="BirkhoffBennett"> <p>Garrett Birkhoff, M. K. Bennett, <em>Felix Klein and His “Erlanger Program”</em>, in Aspray & Kitcher (eds.), <em>History and Philosophy of Modern Mathematics</em>, University of Minnesota Press (1988) 145-176 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.jstor.org/stable/10.5749/j.cttttp0k.9">jstor:10.5749/j.cttttp0k.9</a>, <a href="https://conservancy.umn.edu/bitstream/handle/11299/185660/11_06Birkhoff.pdf?sequence=1">pdf</a>, BirkhoffBennett-KleinAndErlangerProgram.pdf_file<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p>Jeremy Gray, <em>Felix Klein’s Erlangen programme</em>, in I. Grattan-Guinness (ed.), <em>Landmark Writings in Western Mathematics 1640-1940</em>, Elsevier (2005) 544-552 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/B978-0-444-50871-3.X5080-3">doi:10.1016/B978-0-444-50871-3.X5080-3</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p>Vladimir Kisil, <em>Erlangen Programme at Large: An Overview</em> (<a href="http://arxiv.org/abs/1106.1686">arXiv:1106.1686</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a href="http://math.ucr.edu/home/baez/erlangen">webpage on the Erlangen program</a></p> </li> </ul> <h3 id="related_commentary">Related commentary</h3> <ul> <li id="Weyl38"> <p><a class="existingWikiWord" href="/nlab/show/Hermann+Weyl">Hermann Weyl</a>, <em>Symmetry</em>, Journal of the Washington Academy of Sciences, <strong>28</strong> 6 (1938) 253-271 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.jstor.org/stable/24530200">jstor:24530200</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ernst+Cassirer">Ernst Cassirer</a>, <em>The concept of Group and The Theory of Perception</em>, Philosophy and Phenomenological Research <strong>5</strong> 1 (1944) 1-36 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.2307/2102891">doi:10.2307/2102891</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="GruenbaumShephard10"> <p>Branko Grünbaum, G. C. Shephard: <em>A hierarchy of classification methods for patterns</em>, Zeitschrift für Kristallographie - Crystalline Materials, <strong>154</strong> (2010) 163-187 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1524/zkri.1981.154.14.165">doi:10.1524/zkri.1981.154.14.165</a>, <a href="https://sci-hub.hkvisa.net/10.1524/zkri.1981.154.3-4.163">sci-hub:10.1524/zkri.1981.154.3-4.163</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 14, 2022 at 16:05:44. 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