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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="noncommutative_geometry">Noncommutative geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> <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> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> <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> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>)</p> <h2 id="topology">Topology</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantale">quantale</a></li> </ul> <h2 id="smooth_and_riemannian_geometry">Smooth and Riemannian geometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dense+subalgebra">dense subalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+triple">spectral triple</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-spectral+triple">2-spectral triple</a></p> </li> </ul> <h2 id="algebraic_geometry">Algebraic geometry</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+scheme">noncommutative scheme</a></li> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantum+flag+variety">quantum flag variety</a></li> <li><a class="existingWikiWord" href="/nlab/show/quantum+Schubert+cell">quantum Schubert cell</a></li> </ul> <h2 id="homotopy_theory">Homotopy theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operator+algebras">model structure on operator algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+structure+on+C%2A-algebras">homotopical structure on C*-algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>, <a class="existingWikiWord" href="/nlab/show/E-theory">E-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <h2 id="relation_to_physics">Relation to physics</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>, <a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bohr+topos">Bohr topos</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/noncommutative+geometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="index_theory">Index theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></p> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+stable+homotopy+theory">noncommutative stable homotopy theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/genus">genus</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></strong></p> <h2 id="definitions">Definitions</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+module">Hilbert module</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a>, <a class="existingWikiWord" href="/nlab/show/pseudodifferential+operator">pseudodifferential operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symbol+of+a+differential+operator">symbol of a differential operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+operator">elliptic operator</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+complex">elliptic complex</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Spin%5Ec+Dirac+operator">Spin^c Dirac operator</a></li> </ul> </li> </ul> <h2 id="index_theorems">Index theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+index">topological index</a>, <a class="existingWikiWord" href="/nlab/show/analytical+index">analytical index</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Singer+index+theorem">Atiyah-Singer index theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Gauss-Bonnet+theorem">Gauss-Bonnet theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch-Riemann-Roch+theorem">Hirzebruch-Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+analytic+index+theory">global analytic index theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hirzebruch+signature+theorem">Hirzebruch signature theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mishchenko-Fomenko+index+theorem">Mishchenko-Fomenko index theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baum-Connes+conjecture">Baum-Connes conjecture</a></p> </li> </ul> <h2 id="higher_genera">Higher genera</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</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='#definitions'>Definitions</a></li> <ul> <li><a href='#connes_noncommutative_geometry'>Connes’ noncommutative geometry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#history'>History</a></li> <ul> <li><a href='#very_early_sources_and_schools'>Very early sources and schools</a></li> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesNoncommutativeGeometryOfFQHE'>Quantum Hall effect via non-commutative geometry</a></li> <li><a href='#in_stringmtheory'>In string/M-theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Spaces can be characterized by their <a class="existingWikiWord" href="/nlab/show/algebra">algebras</a> of functions (see at <a class="existingWikiWord" href="/nlab/show/Isbell+duality+-+table">Isbell duality - table</a>).</p> <p>Using this <a class="existingWikiWord" href="/nlab/show/duality">duality</a> between <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a> one can define generalized spaces in terms of generalizations of their algebras of functions.</p> <p>The idea of noncommutative geometry is to encode everything about the geometry of a space algebraically and then allow all commutative function algebras to be generalized to possibly non-commutative algebras.</p> <p>More generally, <strong>noncommutative geometry</strong> means replacing the space by some structure carried by an entity (or a collection of entities) living on that would-be space. The entity may be for example a function, vector bundle, coherent sheaf, a complex of sheaves and KK-theory class. Objects organize into <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a>, <a class="existingWikiWord" href="/nlab/show/operator+algebras">operator algebras</a>, <a class="existingWikiWord" href="/nlab/show/categories">categories</a>, higher categories (<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>-linear or not) and so on; and sometimes such a collection represents a space. <a class="existingWikiWord" href="/nlab/show/reconstruction">Reconstruction</a> theorems are theorems on construction of a genuine (say topological) “underlying” space of such an entity or collection. Spectral theories are procedures (sometimes functors, often not) which recover some form of underlying space called <a class="existingWikiWord" href="/nlab/show/spectrum+%28geometry%29">spectrum</a>, often just partially or under strong assumptions on (the data determining) the noncommutative space.</p> <p>Under the process of forming <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebras">groupoid convolution algebras</a> a good bit of commutative but <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> translates into noncommutative geometry. This is for instance the origin of the role of noncommutative geometry in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>.</p> <h2 id="definitions">Definitions</h2> <h3 id="connes_noncommutative_geometry">Connes’ noncommutative geometry</h3> <p>A particular and most prominent realization of the program of noncommutative geometry has been lead by <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>. This is really “<a class="existingWikiWord" href="/nlab/show/spectral+geometry">spectral</a>” and possibly non-commutative <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a>, where the metric structure on a possibly noncommutative “<a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a>” is encoded by a structure called a <em><a class="existingWikiWord" href="/nlab/show/spectral+triple">spectral triple</a></em> (see there for more).</p> <p>The central ingredients in Connes’ noncommutative geometry are</p> <ul> <li> <p>the idea to characterize a (noncommutative) space by a <a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, to be thought of as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of global functions</a> on that space; this approach has been occasionally considered earlier e.g. in the book of Semadeni on Banach algebras.</p> </li> <li> <p>There is a refined, quantized differential calculus where the differential is given by a commutator formula involving a Fredholm operator; the setup in which this is taken place involves cyclic cocycles discovered by Tsygan and Connes.</p> </li> <li> <p>the noncommutative analog of the structure of a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> with a spin structure in terms of generalized <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> acting on a representation space of the algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Metric information on the space is then encoded in the spectrum of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </li> </ul> <p>For that reason <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Connes</a>‘ noncommutative manifolds are well described as <strong>spectral geometry</strong>. The Dirac operator is, of course, very much related to the quantized differential calculus of Connes.</p> <ul> <li>Noncommutative measure spaces are represented by noncommutative von Neumann algebras.</li> </ul> <p>In noncommutative geometry various homotopical and (co)homological invariants were introduced by large amount of improvisation, similar to the beginnings of algebraic topology, but more recently there are few systematic approaches to homotopy theory emerging. See <a class="existingWikiWord" href="/nlab/show/model+structure+on+operator+algebras">model structure on operator algebras</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+measure+theory">noncommutative measure theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+probability+theory">noncommutative probability theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+algebra">groupoid algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+torus">noncommutative torus</a>, <a class="existingWikiWord" href="/nlab/show/fuzzy+sphere">fuzzy sphere</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+topology+of+quasiperiodicity">noncommutative topology of quasiperiodicity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+scheme">noncommutative scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/non-commutative+analytic+space">non-commutative analytic space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+on+non-commutative+spacetime">QFT on non-commutative spacetime</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-brane+geometry">D-brane geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></strong></p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/category">category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dual+category">dual category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a></mtext></mover><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a></mtext></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup><mo>,</mo><mi>comm</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">comm. C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncomm. topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCTopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">NCTopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>general <a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>Schemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}Schemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} </annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A} \phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncomm. algebraic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCSchemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">NCSchemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mrow><mi>fin</mi><mo>,</mo><mi>red</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">fin. gen.</a> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi></mrow><annotation encoding="application/x-tex">SmoothManifolds</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mi>comm</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>SuperSpaces</mi> <mi>Cart</mi></msub></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mphantom><mtext>Pursell's theorem</mtext></mphantom></mover></mtd> <mtd><msubsup><mi>Alg</mi> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom></mrow> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \overset{\phantom{\text{Pursell's theorem}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+moduli+problem">formal</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super Lie theory</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mi>Super</mi><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>fin</mi></msub></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}</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><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mtext><a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a></mtext><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>sdgcAlg</mi> <mi>op</mi></msup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">FDAs</a>”)</td></tr> </tbody></table> <p><strong>in <a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>:</p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/En-algebras">En-algebras</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">BD</a>-<a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</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><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism representation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="history">History</h2> <h3 id="very_early_sources_and_schools">Very early sources and schools</h3> <p>There are many sources of noncommutative spaces, e.g. <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> (Snyder studied an interesting noncommutative space in the late 1940s). It has been often noticed, since Gel’fand–Neimark’s work (see <a class="existingWikiWord" href="/nlab/show/Gelfand+spectrum">Gelfand spectrum</a>), that many geometric notions for commutative <a class="existingWikiWord" href="/nlab/show/Banach+algebra">Banach algebra</a>s are interesting in the noncommutative case; among early enthusiasts one could mention <a class="existingWikiWord" href="/nlab/show/Irving+Segal">Irving Segal</a>; Semadeni’s monograph on Banach spaces of continuous maps also predates the sudden expansion of the field in the late 1970s when <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a> (<a href="http://www.alainconnes.org/">web site</a>) brought about a whole revolution in mathematics using the framework of noncommutative geometry based on <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>s, and boosted by the discovery of cyclic (co)homology by Connes and Boris Tsygan and its connection to <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/ring+theory">ring theory</a>, there were many efforts at building <a class="existingWikiWord" href="/nlab/show/spectrum+%28geometry%29">spectra</a> of noncommutative rings or <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian categories</a> (the spectrum of indecomposable injectives of Gabriel, the affine spectrum of P. M. Cohn and so on) and stronger results in special cases like P. I. rings (polynomial identities rings) by M. Artin, F. van Oystaeyen and others in the 1970s. Golan and van Oystaeyen took noncommutative localization seriously in the mid 1970s with very promising results; <a class="existingWikiWord" href="/nlab/show/Alexander+Rosenberg">A. Rosenberg</a> found in 1982 the so-called <span class="newWikiWord">left spectrum<a href="/nlab/new/left+spectrum">?</a></span> of a noncommutative ring, straightforwardly later generalized to a spectrum of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. All these efforts belong to an early phase of <a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a>. In the late 1980s, the study of <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> of (quasi)coherent sheaves complemented that effort (<a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Kapranov</a>, Bondal, Orlov, …). Though the effort was rather disconnected (from ring-theoretic mainstream noncommutative algebraic geometry) at the beginning, it can be now more easily judged as closely related; this is also the time of birth of the influential school of noncommutative <a class="existingWikiWord" href="/nlab/show/projective+geometry">projective geometry</a> (Artin, Smith, Zhang, …).</p> <p>The quantization program and the study of <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> brought about a number of interesting examples (in the early phase by G. Kac, Sklyanin, Fadeev, Drinfel’d, Woronowicz, Jimbo, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Manin</a>, Reshetikin, Lusztig, Majid and others) with group-like flavour, the study of <a class="existingWikiWord" href="/nlab/show/quantum+groups">quantum groups</a> which were studied in a number of formalisms from operator algebraic to algebro-geometric and purely categorical. <a class="existingWikiWord" href="/nlab/show/action">Actions</a> are very important in noncommutative geometry, and are some of the main examples in Connes’ school like group(oid) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras, crossed product operator algebras and the study of functions on orbifolds and foliations. They also play an important role in <a class="existingWikiWord" href="/nlab/show/equivariant+noncommutative+algebraic+geometry">equivariant noncommutative algebraic geometry</a>; cf. the central notions like <a class="existingWikiWord" href="/nlab/show/Hopf%E2%80%93Galois+extension">Hopf–Galois extension</a> and entwining structure.</p> <p>Noncommutative formal geometry, concerned with objects like infinitesimal neighborhoods of subvariaties, power series in noncommutative variables and so on, has been appearing more sporadically than the operator algebraic and algebro-geometric frameworks. Among pioneering works, there are <a class="existingWikiWord" href="/nlab/show/Kontsevich">Kontsevich</a>‘s (formal) noncommutative symplectic geometry (1992) and <a class="existingWikiWord" href="/nlab/show/Kapranov%27s+noncommutative+geometry">Kapranov's noncommutative geometry</a> “based on commutator expansions”. Noncommutative <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> is even now only vaguely outlined in existing works.</p> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry">Noncommutative Geometry</a></em>, Academic Press, San Diego, CA, 1994 (<a href="https://www.elsevier.com/books/noncommutative-geometry/connes/978-0-08-057175-1">ISBN:9780080571751</a>, <a href="http://www.alainconnes.org/docs/book94bigpdf.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jos%C3%A9+Gracia-Bondia">José Gracia-Bondia</a>, <a class="existingWikiWord" href="/nlab/show/Joseph+Varilly">Joseph Varilly</a>, Héctor Figueroa, <em>Elements of noncommutative geometry</em>, Birkhäuser 2001. xviii+685 pp. (<a href="https://link.springer.com/book/10.1007/978-1-4612-0005-5">doi:10.1007/978-1-4612-0005-5</a>, <a href="http://books.google.hr/books?id=2yJIwWbh1isC&lpg=PP1&ots=ex0Xfmh_UU&dq=Varilly%20noncommutative&hl=en&pg=PP1#v=onepage&q=Varilly%20noncommutative&f=false">gBooks</a>)</p> </li> </ul> <p>Exposition:</p> <ul> <li id="Lupercio20"><a class="existingWikiWord" href="/nlab/show/Ernesto+Lupercio">Ernesto Lupercio</a>, <em>Non-commutative Geometry Indomitable</em> (<a href="https://arxiv.org/abs/2008.08529">arXiv:2008.08529</a>)</li> </ul> <p>See also:</p> <ul> <li id="Connes95"> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em>Noncommutative geometry and reality</em>, J. Math. Phys. 36 (11), 1995 (<a href="http://www.alainconnes.org/docs/reality.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em>A walk in the non-commutative garden</em> (<a href="http://arxiv.org/abs/math/0601054">arXiv:math/0601054</a>)</p> </li> </ul> <p>With a view towards <a class="existingWikiWord" href="/nlab/show/motives+in+physics">motives in physics</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry%2C+Quantum+Fields+and+Motives">Noncommutative Geometry, Quantum Fields and Motives</a></em></li> </ul> <div> <h3 id="ReferencesNoncommutativeGeometryOfFQHE">Quantum Hall effect via non-commutative geometry</h3> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/integer+quantum+Hall+effect">integer quantum Hall effect</a> via a <a class="existingWikiWord" href="/nlab/show/Brillouin+torus">Brillouin torus</a> with <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a> and using the Connes-<a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a>:</p> <ul> <li id="BvESB94">J. Bellissard, A. van Elst, H. Schulz-Baldes: <em>The noncommutative geometry of the quantum Hall effect</em>, J. Math. Phys. <strong>35</strong> 5373 (1994) [<a href="http://link.aip.org/link/doi/10.1063/1.530758">doi:10.1063/1.530758</a>, <a href="http://arxiv.org/abs/cond-mat/9411052">cond-mat/9411052</a>]</li> </ul> <p>Generalization of <a href="#BvESB94">BvESB94</a> to the <a class="existingWikiWord" href="/nlab/show/fractional+quantum+Hall+effect">fractional quantum Hall effect</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>: <em>Towards the fractional quantum Hall effect: a noncommutative geometry perspective</em>, in <em>Noncommutative Geometry and Number Theory</em>, Vieweg (2006) [<a href="https://doi.org/10.1007/978-3-8348-0352-8_12">doi:10.1007/978-3-8348-0352-8_12</a>, <a href="https://arxiv.org/abs/cond-mat/0502356">arXiv:cond-mat/0502356</a>]</li> </ul> <p>See also exposition in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Vincent+Pasquier">Vincent Pasquier</a>: <em>Quantum Hall Effect and Non-commutative Geometry</em>, in: <em>Quantum Spaces</em>, Progress in Mathematical Physics <strong>53</strong>, Birkhäuser (2007) [<a href="https://doi.org/10.1007/978-3-7643-8522-4_1">doi:10.1007/978-3-7643-8522-4_1</a>, <a href="http://www.bourbaphy.fr/pasquier.pdf">pdf</a>]</li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/fractional+quantum+Hall+effect">fractional quantum Hall effect</a> via <a class="existingWikiWord" href="/nlab/show/abelian+Chern-Simons+theory">abelian</a> but <a class="existingWikiWord" href="/nlab/show/noncommutative+field+theory">noncommutative</a> (<a class="existingWikiWord" href="/nlab/show/matrix+model">matrix model</a>-)<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leonard+Susskind">Leonard Susskind</a>: <em>The Quantum Hall Fluid and Non-Commutative Chern Simons Theory</em> [<a href="https://arxiv.org/abs/hep-th/0101029">arXiv:hep-th/0101029</a>]</p> </li> <li id="HellermanSusskind01"> <p><a class="existingWikiWord" href="/nlab/show/Simeon+Hellerman">Simeon Hellerman</a>, <a class="existingWikiWord" href="/nlab/show/Leonard+Susskind">Leonard Susskind</a>: <em>Realizing the Quantum Hall System in String Theory</em> [<a href="https://arxiv.org/abs/hep-th/0107200">arXiv:hep-th/0107200</a>]</p> <blockquote> <p>(relating this to <a class="existingWikiWord" href="/nlab/show/M5-branes">M5-branes</a> via the <a class="existingWikiWord" href="/nlab/show/BFSS+matrix+model">BFSS matrix model</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexios+P.+Polychronakos">Alexios P. Polychronakos</a>: <em>Quantum Hall states as matrix Chern-Simons theory</em>, JHEP 0104:011 (2001) [<a href="https://doi.org/10.1088/1126-6708/2001/04/011">doi:10.1088/1126-6708/2001/04/011</a>, <a href="https://arxiv.org/abs/hep-th/0103013">arXiv:hep-th/0103013</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Simeon+Hellerman">Simeon Hellerman</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Van+Raamsdonk">Mark Van Raamsdonk</a>: <em>Quantum Hall Physics = Noncommutative Field Theory</em>, JHEP 0110:039 (2001) [<a href="https://doi.org/10.1088/1126-6708/2001/10/039">doi:10.1088/1126-6708/2001/10/039</a>, <a href="https://arxiv.org/abs/hep-th/0103179">arXiv:hep-th/0103179</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eduardo+Fradkin">Eduardo Fradkin</a>, Vishnu Jejjala, <a class="existingWikiWord" href="/nlab/show/Robert+G.+Leigh">Robert G. Leigh</a>: <em>Non-commutative Chern-Simons for the Quantum Hall System and Duality</em>, Nucl. Phys. B <strong>642</strong> (2002) 483-500 [<a href="https://doi.org/10.1016/S0550-3213(02)00616-8">doi:10.1016/S0550-3213(02)00616-8</a>, <a href="https://arxiv.org/abs/cond-mat/0205653">arXiv:cond-mat/0205653</a>]</p> </li> <li> <p>Andrea Cappelli, Ivan D. Rodriguez: <em>Matrix Effective Theories of the Fractional Quantum Hall effect</em>, J. Phys. A <strong>42</strong> (2009) 304006 [<a href="https://doi.org/10.1088/1751-8113/42/30/304006">doi:10.1088/1751-8113/42/30/304006</a>, <a href="https://arxiv.org/abs/0902.0765">arXiv:0902.0765</a>]</p> </li> <li> <p>Zhihuan Dong, T. Senthil: <em>Non-commutative field theory and composite Fermi Liquids in some quantum Hall systems</em>, Phys. Rev. B <strong>102</strong> (2020) 205126 [<a href="https://doi.org/10.1103/PhysRevB.102.205126">doi:10.1103/PhysRevB.102.205126</a>, <a href="https://arxiv.org/abs/2006.01282">arXiv:2006.01282</a>]</p> </li> </ul> </div> <h3 id="in_stringmtheory">In string/M-theory</h3> <p>Discussion of effective non-commutative geometry on <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> induced from a constant <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> <a class="existingWikiWord" href="/nlab/show/flux+density">flux density</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/David+S.+Berman">David S. Berman</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Pieter+van+der+Schaar">Jan Pieter van der Schaar</a>, <a class="existingWikiWord" href="/nlab/show/Per+Sundell">Per Sundell</a>: <em>A Noncommutative M-Theory Five-brane</em>, Nucl. Phys. B <strong>590</strong> (2000) 173-197 [<a href="https://doi.org/10.1016/S0550-3213(00)00476-4">doi:10.1016/S0550-3213(00)00476-4</a>, <a href="https://arxiv.org/abs/hep-th/0005026">arXiv:hep-th/0005026</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Bergshoeff">Eric Bergshoeff</a>, <a class="existingWikiWord" href="/nlab/show/David+S.+Berman">David S. Berman</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Pieter+van+der+Schaar">Jan Pieter van der Schaar</a>, <a class="existingWikiWord" href="/nlab/show/Per+Sundell">Per Sundell</a>: <em>Critical fields on the M5-brane and noncommutative open strings</em>, Phys. Lett. B <strong>492</strong> (2000) 193-200 [<a href="https://doi.org/10.1016/S0370-2693(00)01081-9">doi:10.1016/S0370-2693(00)01081-9</a>, <a href="https://arxiv.org/abs/hep-th/0006112">arXiv:hep-th/0006112</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Rajesh+Gopakumar">Rajesh Gopakumar</a>, <a class="existingWikiWord" href="/nlab/show/Shiraz+Minwalla">Shiraz Minwalla</a>, <a class="existingWikiWord" href="/nlab/show/Nathan+Seiberg">Nathan Seiberg</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Strominger">Andrew Strominger</a>: <em>OM Theory in Diverse Dimensions</em>, JHEP08 (2000) 008 [<a href="https://iopscience.iop.org/article/10.1088/1126-6708/2000/08/008">doi:10.1088/1126-6708/2000/08/008</a>, <a href="https://arxiv.org/abs/hep-th/0006062">arXiv:hep-th/0006062</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Micha+Berkooz">Micha Berkooz</a>: <em>Light-like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> Noncommutativity and Light-Cone Rigid Open Membrane Theory</em> [<a href="https://arxiv.org/abs/hep-th/0010158">arXiv:hep-th/0010158</a>, <a href="https://inspirehep.net/literature/535327">inspire:535327</a>]</p> </li> </ul> <p>and relating this to aspects of the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall effect</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Simeon+Hellerman">Simeon Hellerman</a>, <a class="existingWikiWord" href="/nlab/show/Leonard+Susskind">Leonard Susskind</a>: <em>Realizing the Quantum Hall System in String Theory</em> [<a href="https://arxiv.org/abs/hep-th/0107200">arXiv:hep-th/0107200</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 17, 2024 at 19:21:45. See the <a href="/nlab/history/noncommutative+geometry" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/noncommutative+geometry" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/10835/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/noncommutative+geometry/42" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/noncommutative+geometry" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/noncommutative+geometry" accesskey="S" class="navlink" id="history" rel="nofollow">History (42 revisions)</a> <a href="/nlab/show/noncommutative+geometry/cite" style="color: black">Cite</a> <a href="/nlab/print/noncommutative+geometry" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/noncommutative+geometry" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>