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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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#RelationToCosetSpaces'>Relation to coset spaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> or <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> or <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, etc., <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>, a <strong>homogeneous <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>-space</strong> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> or <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>, or <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> etc. with <a class="existingWikiWord" href="/nlab/show/transitive+action">transitive</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>-<a class="existingWikiWord" href="/nlab/show/action">action</a>.</p> <p>A <strong>principal homogeneous <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>-space</strong> is the total space of 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>-<a class="existingWikiWord" href="/nlab/show/torsor">torsor</a> over a point.</p> <p>There are generalizations, e.g. the <a class="existingWikiWord" href="/nlab/show/quantum+homogeneous+space">quantum homogeneous space</a> for the case of quantum groups.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>A special case of homogeneous spaces are <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a> arising from the <a class="existingWikiWord" href="/nlab/show/quotient">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><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/group">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> by a subgroup. For the case of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> this is also called <a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>.</p> </li> <li> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">T\hookrightarrow G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/maximal+torus">maximal torus</a>, then the coset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G/T</annotation></semantics></math> play a central role in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, for instance in the <a class="existingWikiWord" href="/nlab/show/splitting+principle">splitting principle</a>.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a> and <a class="existingWikiWord" href="/nlab/show/number+theory">number theory</a>, certain functions on certain coset spaces play a role as <em><a class="existingWikiWord" href="/nlab/show/automorphic+forms">automorphic forms</a></em> (e.g. <a class="existingWikiWord" href="/nlab/show/modular+forms">modular forms</a>). See there for more.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="RelationToCosetSpaces">Relation to coset spaces</h3> <p>Under weak topological conditions (cf. <a href="#Helgason">Helgason</a>), every topological homogeneous space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is isomorphic to a <strong><a class="existingWikiWord" href="/nlab/show/coset">coset</a> space</strong> <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> for a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> subgroup <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\subset G</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/stabilizer">stabilizer</a> of a fixed point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conjugacy+class">conjugacy class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grassmannian">Grassmannian</a>, <a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schubert+calculus">Schubert calculus</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li id="Bredon72"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Section I.4 of: <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+compact+transformation+groups">Introduction to compact transformation groups</a></em>, Academic Press 1972 (<a href="https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1">ISBN 9780080873596</a>, <a href="http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf">pdf</a>)</p> </li> <li id="Helgason"> <p>Sigurdur Helgason, <em>Differential geometry, Lie groups and symmetric spaces</em></p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/homogeneous+spaces">homogeneous spaces</a> with the same <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> as a <a class="existingWikiWord" href="/nlab/show/product+space">product</a> of <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Linus+Kramer">Linus Kramer</a>, <em>Homogeneous Spaces, Tits Buildings, and Isoparametric Hypersurface</em>, Memoirs of the American Mathematical Society number 752 (<a href="http://arxiv.org/abs/math/0109133">arXiv:math/0109133</a>, <a href="http://dx.doi.org/10.1090/memo/0752">doi:10.1090/memo/0752</a>, <a href="http://books.google.com/books?id=SA8O6ihrDFkC&printsec=frontcover&hl=de&source=gbs_v2_summary_r&cad=0#v=onepage&q=&f=false">GoogleBooks</a>)</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> analysis of relation between the total space of a principal bundle and of the corresponding quotient space both for the classical case and for <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative</a> generalizations:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tomasz+Brzezi%C5%84ski">Tomasz Brzeziński</a>, <em>On synthetic interpretation of quantum principal bundles</em>, AJSE D - Mathematics <strong>35</strong> 1D (2010) 13-27 [<a href="http://uk.arxiv.org/abs/0912.0213">arxiv:0912.0213</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tomasz+Brzezi%C5%84ski">Tomasz Brzeziński</a>, <em>Quantum group differentials, bundles and gauge theory</em>, Encyclopedia of Mathematical Physics, Acad. Press. (2006) 236-244 [<a href="http://dx.doi.org/10.1016/B0-12-512666-2/00050-X">doi:10.1016/B0-12-512666-2/00050-X</a>]</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a> (<a class="existingWikiWord" href="/nlab/show/homogeneous+spaces">homogeneous spaces</a>) and their <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+forms">Maurer-Cartan forms</a> in application to <a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation</a> of (<a class="existingWikiWord" href="/nlab/show/supergravity">super</a>-)<a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/L.+J.+Romans">L. J. Romans</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+P.+Warner">Nicholas P. Warner</a>, <em>Symmetries of coset spaces and Kaluza-Klein supergravity</em>, Annals of Physics <strong>157</strong> 2 (1984) 394-407 [<a href="https://doi.org/10.1016/0003-4916(84)90066-6">doi:10.1016/0003-4916(84)90066-6</a>]</p> </li> <li id="CastellaniDAuriaFre"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <a class="existingWikiWord" href="/nlab/show/Riccardo+D%27Auria">Riccardo D'Auria</a>, <a class="existingWikiWord" href="/nlab/show/Pietro+Fr%C3%A9">Pietro Fré</a>, §I.6 in: <em><a class="existingWikiWord" href="/nlab/show/Supergravity+and+Superstrings+-+A+Geometric+Perspective">Supergravity and Superstrings - A Geometric Perspective</a></em>, World Scientific (1991) [<a href="https://doi.org/10.1142/0224">doi:10.1142/0224</a>, toc: <a class="existingWikiWord" href="/nlab/files/CDF91-TOC.pdf" title="pdf">pdf</a>, ch I.6: <a class="existingWikiWord" href="/nlab/files/CastellaniDAuriaFre-ChI6.pdf" title="pdf">pdf</a></p> </li> <li id="Castellani01"> <p><a class="existingWikiWord" href="/nlab/show/Leonardo+Castellani">Leonardo Castellani</a>, <em>On G/H geometry and its use in M-theory compactifications</em>, Annals Phys. <strong>287</strong> (2001) 1-13 [<a href="https://arxiv.org/abs/hep-th/9912277">arXiv:hep-th/9912277</a>, <a href="https://doi.org/10.1006/aphy.2000.6097">doi:10.1006/aphy.2000.6097</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 25, 2024 at 21:10:20. 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