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<span class="vector-toc-numb">2</span> <span>Binary polyhedral groups</span> </div> </a> <ul id="toc-Binary_polyhedral_groups-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Labeled_graphs" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Labeled_graphs"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Labeled graphs</span> </div> </a> <ul id="toc-Labeled_graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_classifications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Other_classifications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Other classifications</span> </div> </a> <ul id="toc-Other_classifications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_of_the_classification" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extension_of_the_classification"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extension of the classification</span> </div> </a> <ul id="toc-Extension_of_the_classification-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Trinities" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Trinities"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Trinities</span> </div> </a> <ul id="toc-Trinities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div 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src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Simply_Laced_Dynkin_Diagrams.svg/220px-Simply_Laced_Dynkin_Diagrams.svg.png" decoding="async" width="220" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Simply_Laced_Dynkin_Diagrams.svg/330px-Simply_Laced_Dynkin_Diagrams.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Simply_Laced_Dynkin_Diagrams.svg/440px-Simply_Laced_Dynkin_Diagrams.svg.png 2x" data-file-width="400" data-file-height="480" /></a><figcaption>The <a href="/wiki/Simply_laced_Dynkin_diagram" class="mw-redirect" title="Simply laced Dynkin diagram">simply laced Dynkin diagrams</a> classify diverse mathematical objects.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>ADE classification</b> (originally <b><i>A-D-E</i> classifications</b>) is a situation where certain kinds of objects are in correspondence with <a href="/wiki/Simply_laced_Dynkin_diagram" class="mw-redirect" title="Simply laced Dynkin diagram">simply laced Dynkin diagrams</a>. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in (<a href="#CITEREFArnold1976">Arnold 1976</a>). The complete list of <a href="/wiki/Simply_laced_Dynkin_diagram" class="mw-redirect" title="Simply laced Dynkin diagram">simply laced Dynkin diagrams</a> comprises </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n},\,D_{n},\,E_{6},\,E_{7},\,E_{8}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n},\,D_{n},\,E_{6},\,E_{7},\,E_{8}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/805f2dedc16eab31f6858af8d14ef678e3ac1614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.744ex; height:2.509ex;" alt="{\displaystyle A_{n},\,D_{n},\,E_{6},\,E_{7},\,E_{8}.}"></span></dd></dl> <p>Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the <a href="/wiki/Root_system" title="Root system">root system</a> forming angles of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi /2=90^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>=</mo> <msup> <mn>90</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi /2=90^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace87a7c23cafe1c0829af5e6011a024ee9a4fcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.135ex; height:2.843ex;" alt="{\displaystyle \pi /2=90^{\circ }}"></span> (no edge between the vertices) or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi /3=120^{\circ }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> <mo>=</mo> <msup> <mn>120</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>∘</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\pi /3=120^{\circ }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bccd8023d96994d1c853189c24bc082720c70c2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.459ex; height:2.843ex;" alt="{\displaystyle 2\pi /3=120^{\circ }}"></span> (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f568bf6d34e97b9fdda0dc7e276d6c4501d2045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.982ex; height:2.509ex;" alt="{\displaystyle B_{n}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0301812adb392070d834ca2df4ed97f1cf132f33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.88ex; height:2.509ex;" alt="{\displaystyle C_{n}}"></span>), and three of the five exceptional Dynkin diagrams (omitting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8718a2df1e70bea3cd21ab9e0cd45dc354818451" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.549ex; height:2.509ex;" alt="{\displaystyle F_{4}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645011b0c6933a02f5f7d84624f78220d747427e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{2}}"></span>). </p><p>This list is non-redundant if one takes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≥</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\geq 4}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25010fec4b0f68f1b46f49d14917d962acca0b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 4}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93d4ed899749eb2323c226dd59cbc09586507f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.79ex; height:2.509ex;" alt="{\displaystyle D_{n}.}"></span> If one extends the families to include redundant terms, one obtains the <a href="/wiki/Exceptional_isomorphism" title="Exceptional isomorphism">exceptional isomorphisms</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>≅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>≅</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>≅</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a32019f7e437d76d1db93a85b70802f9162f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.101ex; height:2.509ex;" alt="{\displaystyle D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},}"></span></dd></dl> <p>and corresponding isomorphisms of classified objects. </p><p>The <i>A</i>, <i>D</i>, <i>E</i> nomenclature also yields the simply laced <a href="/wiki/Finite_Coxeter_group" class="mw-redirect" title="Finite Coxeter group">finite Coxeter groups</a>, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Lie_algebras">Lie algebras</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=1" title="Edit section: Lie algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In terms of complex semisimple Lie algebras: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span> corresponds to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {sl}}_{n+1}(\mathbf {C} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">l</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {sl}}_{n+1}(\mathbf {C} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd45fe0846ca66f4484616f134fca065d5edd7be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:9.429ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {sl}}_{n+1}(\mathbf {C} ),}"></span> the <a href="/wiki/Special_linear_Lie_algebra" title="Special linear Lie algebra">special linear Lie algebra</a> of <a href="/wiki/Traceless" class="mw-redirect" title="Traceless">traceless</a> operators,</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe03857347bf517e7fbda4085b0dafd6018cf18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle D_{n}}"></span> corresponds to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}_{2n}(\mathbf {C} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}_{2n}(\mathbf {C} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65c8db1b8d13ba035343d02a5f421d317b8ffb4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:8.635ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}_{2n}(\mathbf {C} ),}"></span> the even <a href="/wiki/Special_orthogonal_Lie_algebra" class="mw-redirect" title="Special orthogonal Lie algebra">special orthogonal Lie algebra</a> of even-dimensional <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric</a> operators, and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{6},E_{7},E_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{6},E_{7},E_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa381a152b7cca1c12854a81bbb40006cef36bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.376ex; height:2.509ex;" alt="{\displaystyle E_{6},E_{7},E_{8}}"></span> are three of the five exceptional Lie algebras.</li></ul> <p>In terms of <a href="/wiki/Compact_Lie_algebra" title="Compact Lie algebra">compact Lie algebras</a> and corresponding <a href="/wiki/Simple_Lie_group#Simply_laced_groups" title="Simple Lie group">simply laced Lie groups</a>: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f6906700685b6d52f3958b1c2ae659d2d97d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.962ex; height:2.509ex;" alt="{\displaystyle A_{n}}"></span> corresponds to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {su}}_{n+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}_{n+1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dedcb7bdb1093ead4b9eb8038b6ac0234c47bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.041ex; width:6.238ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {su}}_{n+1},}"></span> the algebra of the <a href="/wiki/Special_unitary_group" title="Special unitary group">special unitary group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(n+1);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(n+1);}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23f27ac7c719344f65b43c2f821456441ab543c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.136ex; height:2.843ex;" alt="{\displaystyle SU(n+1);}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe03857347bf517e7fbda4085b0dafd6018cf18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.143ex; height:2.509ex;" alt="{\displaystyle D_{n}}"></span> corresponds to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {so}}_{2n}(\mathbf {R} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">o</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {so}}_{2n}(\mathbf {R} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cffc91fc571a554f2db16308605c9b6699ae88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:8.707ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {so}}_{2n}(\mathbf {R} ),}"></span> the algebra of the even <a href="/wiki/Projective_special_orthogonal_group" class="mw-redirect" title="Projective special orthogonal group">projective special orthogonal group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle PSO(2n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mi>S</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle PSO(2n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd0bf53088c050af70c9f8560fa776ac0c62db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.384ex; height:2.843ex;" alt="{\displaystyle PSO(2n)}"></span>, while</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{6},E_{7},E_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{6},E_{7},E_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aa381a152b7cca1c12854a81bbb40006cef36bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.376ex; height:2.509ex;" alt="{\displaystyle E_{6},E_{7},E_{8}}"></span> are three of five exceptional <a href="/wiki/Compact_Lie_algebra" title="Compact Lie algebra">compact Lie algebras</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Binary_polyhedral_groups">Binary polyhedral groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=2" title="Edit section: Binary polyhedral groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The same classification applies to discrete subgroups of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f8cd5de228a45abf34210c1666cd46dd87bc12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.254ex; height:2.843ex;" alt="{\displaystyle SU(2)}"></span>, the <a href="/wiki/Binary_polyhedral_group" class="mw-redirect" title="Binary polyhedral group">binary polyhedral groups</a>; properly, binary polyhedral groups correspond to the simply laced <i>affine</i> <a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagrams</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {A}}_{n},{\tilde {D}}_{n},{\tilde {E}}_{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {A}}_{n},{\tilde {D}}_{n},{\tilde {E}}_{k},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6b1ec0c10b5e4365438d14acacdad4c36ac4ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.716ex; height:3.009ex;" alt="{\displaystyle {\tilde {A}}_{n},{\tilde {D}}_{n},{\tilde {E}}_{k},}"></span> and the representations of these groups can be understood in terms of these diagrams. This connection is known as the <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="McKay_correspondence"></span><span class="vanchor-text">McKay correspondence</span></span></b> after <a href="/wiki/John_McKay_(mathematician)" title="John McKay (mathematician)">John McKay</a>. The connection to <a href="/wiki/Platonic_solid" title="Platonic solid">Platonic solids</a> is described in (<a href="#CITEREFDickson1959">Dickson 1959</a>). The correspondence uses the construction of <a href="/wiki/McKay_graph" title="McKay graph">McKay graph</a>. </p><p>Note that the ADE correspondence is <i>not</i> the correspondence of Platonic solids to their <a href="/wiki/Reflection_group" title="Reflection group">reflection group</a> of symmetries: for instance, in the ADE correspondence the <a href="/wiki/Tetrahedron" title="Tetrahedron">tetrahedron</a>, <a href="/wiki/Cube" title="Cube">cube</a>/<a href="/wiki/Octahedron" title="Octahedron">octahedron</a>, and <a href="/wiki/Dodecahedron" title="Dodecahedron">dodecahedron</a>/<a href="/wiki/Icosahedron" title="Icosahedron">icosahedron</a> correspond to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{6},E_{7},E_{8},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{6},E_{7},E_{8},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df54f94167db4d3c3740829001e2d46414191dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.023ex; height:2.509ex;" alt="{\displaystyle E_{6},E_{7},E_{8},}"></span> while the reflection groups of the tetrahedron, cube/octahedron, and dodecahedron/icosahedron are instead representations of the <a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter groups</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{3},BC_{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mi>B</mi> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{3},BC_{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fac20eb25b83b762a8b06360a90e6d690b4ab0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.958ex; height:2.509ex;" alt="{\displaystyle A_{3},BC_{3},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{3}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f101664f6aaf86902f5d1efb0fc9f92caf7d83fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.632ex; height:2.509ex;" alt="{\displaystyle H_{3}.}"></span> </p><p>The <a href="/wiki/Orbifold" title="Orbifold">orbifold</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b977492b76f197efebf8bceb399814966be79c35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.985ex; height:2.676ex;" alt="{\displaystyle \mathbf {C} ^{2}}"></span> constructed using each discrete subgroup leads to an ADE-type singularity at the origin, termed a <a href="/wiki/Du_Val_singularity" title="Du Val singularity">du Val singularity</a>. </p><p>The McKay correspondence can be extended to multiply laced Dynkin diagrams, by using a <i>pair</i> of binary polyhedral groups. This is known as the <b>Slodowy correspondence</b>, named after <a href="/wiki/Peter_Slodowy" title="Peter Slodowy">Peter Slodowy</a> – see (<a href="#CITEREFStekolshchik2008">Stekolshchik 2008</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Labeled_graphs">Labeled graphs</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=3" title="Edit section: Labeled graphs"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The ADE graphs and the extended (affine) ADE graphs can also be characterized in terms of labellings with certain properties,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> which can be stated in terms of the <a href="/wiki/Discrete_Laplace_operator" title="Discrete Laplace operator">discrete Laplace operators</a><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Cartan_matrices" class="mw-redirect" title="Cartan matrices">Cartan matrices</a>. Proofs in terms of Cartan matrices may be found in (<a href="#CITEREFKac1990">Kac 1990</a>, pp. 47–54). </p><p>The affine ADE graphs are the only graphs that admit a positive labeling (labeling of the nodes by positive real numbers) with the following property: </p> <dl><dd>Twice any label is the sum of the labels on adjacent vertices.</dd></dl> <p>That is, they are the only positive functions with eigenvalue 1 for the discrete Laplacian (sum of adjacent vertices minus value of vertex) – the positive solutions to the homogeneous equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \phi =\phi .\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>=</mo> <mi>ϕ</mi> <mo>.</mo> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \phi =\phi .\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0079325c39e7e33a69bfd4a02e4e1b79a62eca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.42ex; height:2.509ex;" alt="{\displaystyle \Delta \phi =\phi .\ }"></span></dd></dl> <p>Equivalently, the positive functions in the kernel of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta -I.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>−</mo> <mi>I</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta -I.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68eecd5e2a162336e1df2895061373aae80a086d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.595ex; height:2.343ex;" alt="{\displaystyle \Delta -I.}"></span> The resulting numbering is unique up to scale, and if normalized such that the smallest number is 1, consists of small integers – 1 through 6, depending on the graph. </p><p>The ordinary ADE graphs are the only graphs that admit a positive labeling with the following property: </p> <dl><dd>Twice any label minus two is the sum of the labels on adjacent vertices.</dd></dl> <p>In terms of the Laplacian, the positive solutions to the inhomogeneous equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \phi =\phi -2.\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ϕ</mi> <mo>=</mo> <mi>ϕ</mi> <mo>−</mo> <mn>2.</mn> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \phi =\phi -2.\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c16dbe7786c8c98f693ea21ca38caec5a6200d2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.036ex; height:2.509ex;" alt="{\displaystyle \Delta \phi =\phi -2.\ }"></span></dd></dl> <p>The resulting numbering is unique (scale is specified by the "2") and consists of integers; for E<sub>8</sub> they range from 58 to 270, and have been observed as early as (<a href="#CITEREFBourbaki1968">Bourbaki 1968</a>). </p> <div class="mw-heading mw-heading2"><h2 id="Other_classifications">Other classifications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=4" title="Edit section: Other classifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Catastrophe_theory#Elementary_catastrophes" title="Catastrophe theory">elementary catastrophes</a> are also classified by the ADE classification. </p><p>The ADE diagrams are exactly the <a href="/wiki/Quiver_(mathematics)" title="Quiver (mathematics)">quivers</a> of finite type, via <a href="/wiki/Gabriel%27s_theorem" title="Gabriel's theorem">Gabriel's theorem</a>. </p><p>There is also a link with <a href="/wiki/Generalized_quadrangle" title="Generalized quadrangle">generalized quadrangles</a>, as the three non-degenerate GQs with three points on each line correspond to the three exceptional root systems <i>E</i><sub>6</sub>, <i>E</i><sub>7</sub> and <i>E</i><sub>8</sub>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> The classes <i>A</i> and <i>D</i> correspond degenerate cases where the line set is empty or we have all lines passing through a fixed point, respectively.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>It was suggested that symmetries of small <a href="/wiki/Droplet_cluster" title="Droplet cluster">droplet clusters</a> may be subject to an ADE classification.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Minimal_model_(physics)" title="Minimal model (physics)">minimal models</a> of <a href="/wiki/Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">two-dimensional conformal field theory</a> have an ADE classification. </p><p>Four dimensional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {N}}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">N</mi> </mrow> </mrow> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {N}}=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81f52d38c18106a1322d74903137ab9a0c87b4d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; margin-left: -0.062ex; width:6.597ex; height:2.509ex;" alt="{\displaystyle {\mathcal {N}}=2}"></span> superconformal gauge quiver theories with unitary gauge groups have an ADE classification. </p> <div class="mw-heading mw-heading2"><h2 id="Extension_of_the_classification">Extension of the classification</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=5" title="Edit section: Extension of the classification"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold</a> has subsequently proposed many further extensions in this classification scheme, in the idea to revisit and generalize the <a href="/wiki/Coxeter-Dynkin_diagram" class="mw-redirect" title="Coxeter-Dynkin diagram">Coxeter classification</a> and <a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin classification</a> under the single umbrella of <a href="/wiki/Root_systems" class="mw-redirect" title="Root systems">root systems</a>. He tried to introduce informal concepts of Complexification and Symplectization based on analogies between <a href="/wiki/Picard%E2%80%93Lefschetz_theory" title="Picard–Lefschetz theory">Picard–Lefschetz theory</a> which he interprets as the Complexified version of <a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a> and then extend them to other areas of mathematics. He tries also to identify hierarchies and dictionaries between mathematical objects and theories where for example <a href="/wiki/Diffeomorphism" title="Diffeomorphism">diffeomorphism</a> corresponds to the A type of the <a href="/wiki/Dynkin_diagrams" class="mw-redirect" title="Dynkin diagrams">Dynkyn classification</a>, volume preserving diffeomorphism corresponds to B type and <a href="/wiki/Symplectomorphisms" class="mw-redirect" title="Symplectomorphisms">Symplectomorphisms</a> corresponds to C type. In the same spirit he revisits analogies between different mathematical objects where for example the <a href="/wiki/Lie_bracket" class="mw-redirect" title="Lie bracket">Lie bracket</a> in the scope of <a href="/wiki/Diffeomorphisms" class="mw-redirect" title="Diffeomorphisms">Diffeomorphisms</a> becomes analogous (and at the same time includes as a special case) the <a href="/wiki/Poisson_bracket" title="Poisson bracket">Poisson bracket</a> of <a href="/wiki/Symplectomorphism" title="Symplectomorphism">Symplectomorphism</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Trinities">Trinities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=6" title="Edit section: Trinities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Arnold extended this further under the rubric of "mathematical trinities".<sup id="cite_ref-arntrin_8-0" class="reference"><a href="#cite_note-arntrin-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> McKay has extended his correspondence along parallel and sometimes overlapping lines. Arnold terms these "<a href="/wiki/Trinity" title="Trinity">trinities</a>" to evoke religion, and suggest that (currently) these parallels rely more on faith than on rigorous proof, though some parallels are elaborated. Further trinities have been suggested by other authors.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-arntrin_8-1" class="reference"><a href="#cite_note-arntrin-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Arnold's trinities begin with <b>R</b>/<b>C</b>/<b>H</b> (the real numbers, complex numbers, and quaternions), which he remarks "everyone knows", and proceeds to imagine the other trinities as "complexifications" and "quaternionifications" of classical (real) mathematics, by analogy with finding symplectic analogs of classic Riemannian geometry, which he had previously proposed in the 1970s. In addition to examples from differential topology (such as <a href="/wiki/Characteristic_class" title="Characteristic class">characteristic classes</a>), Arnold considers the three Platonic symmetries (tetrahedral, octahedral, icosahedral) as corresponding to the reals, complexes, and quaternions, which then connects with McKay's more algebraic correspondences, below. </p><p><a href="/wiki/McKay_correspondence" class="mw-redirect" title="McKay correspondence">McKay's correspondences</a> are easier to describe. Firstly, the extended Dynkin diagrams <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9372d1383c3b009ace8d788984e00882ded92a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.557ex; height:3.009ex;" alt="{\displaystyle {\tilde {E}}_{6},{\tilde {E}}_{7},{\tilde {E}}_{8}}"></span> (corresponding to tetrahedral, octahedral, and icosahedral symmetry) have symmetry groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{3},S_{2},S_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{3},S_{2},S_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab485a9a0a71aff7f19407978a29fdefe9c8ac54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.152ex; height:2.509ex;" alt="{\displaystyle S_{3},S_{2},S_{1},}"></span> respectively, and the associated <a href="/wiki/Folding_(Dynkin_diagram)" class="mw-redirect" title="Folding (Dynkin diagram)">foldings</a> are the diagrams <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {G}}_{2},{\tilde {F}}_{4},{\tilde {E}}_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>G</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {G}}_{2},{\tilde {F}}_{4},{\tilde {E}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25831abdb28421cee3914e8ea15c5acc8a2c7364" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.638ex; height:3.009ex;" alt="{\displaystyle {\tilde {G}}_{2},{\tilde {F}}_{4},{\tilde {E}}_{8}}"></span> (note that in less careful writing, the extended (tilde) qualifier is often omitted). More significantly, McKay suggests a correspondence between the nodes of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {E}}_{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7960ec54a7dac08a847e38ee3137a3e95a9044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:3.009ex;" alt="{\displaystyle {\tilde {E}}_{8}}"></span> diagram and certain conjugacy classes of the <a href="/wiki/Monster_group" title="Monster group">monster group</a>, which is known as <i>McKay's E<sub>8</sub> observation;</i><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-monster_12-0" class="reference"><a href="#cite_note-monster-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> see also <a href="/wiki/Monstrous_moonshine" title="Monstrous moonshine">monstrous moonshine</a>. McKay further relates the nodes of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {E}}_{7}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c251975c254e8e74513a013145d6ec2b0e5d016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:3.009ex;" alt="{\displaystyle {\tilde {E}}_{7}}"></span> to conjugacy classes in 2.<i>B</i> (an order 2 extension of the <a href="/wiki/Baby_monster_group" title="Baby monster group">baby monster group</a>), and the nodes of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {E}}_{6}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tilde {E}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6f88e9053912a19c4fc76340931dd573140944e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:3.009ex;" alt="{\displaystyle {\tilde {E}}_{6}}"></span> to conjugacy classes in 3.<i>Fi</i><sub>24</sub>' (an order 3 extension of the <a href="/wiki/Fischer_group" title="Fischer group">Fischer group</a>)<sup id="cite_ref-monster_12-1" class="reference"><a href="#cite_note-monster-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> – note that these are the three largest <a href="/wiki/Sporadic_group" title="Sporadic group">sporadic groups</a>, and that the order of the extension corresponds to the symmetries of the diagram. </p><p>Turning from large simple groups to small ones, the corresponding Platonic groups <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{4},S_{4},A_{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{4},S_{4},A_{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/878239846301b9cac22ba6f9468591cc13642ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.142ex; height:2.509ex;" alt="{\displaystyle A_{4},S_{4},A_{5}}"></span> have connections with the <a href="/wiki/Projective_special_linear_group" class="mw-redirect" title="Projective special linear group">projective special linear groups</a> PSL(2,5), PSL(2,7), and PSL(2,11) (orders 60, 168, and 660),<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> which is deemed a "McKay correspondence".<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> These groups are the only (simple) values for <i>p</i> such that PSL(2,<i>p</i>) <a href="/wiki/Projective_linear_group#Action_on_p_points" title="Projective linear group">acts non-trivially on <i>p</i> points</a>, a fact dating back to <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Évariste Galois</a> in the 1830s. In fact, the groups decompose as products of sets (not as products of groups) as: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{4}\times Z_{5},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{4}\times Z_{5},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6512109eab2bbe1fa997ae55dfa6520b5b3c5c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.926ex; height:2.509ex;" alt="{\displaystyle A_{4}\times Z_{5},}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{4}\times Z_{7},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{4}\times Z_{7},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc76c79d16568e65dab939aaa090f4b96d1344f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.608ex; height:2.509ex;" alt="{\displaystyle S_{4}\times Z_{7},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{5}\times Z_{11}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> <mo>×</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{5}\times Z_{11}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1119e08fbafa21f85a8a7b30b78f82445e9c2af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.748ex; height:2.509ex;" alt="{\displaystyle A_{5}\times Z_{11}.}"></span> These groups also are related to various geometries, which dates to <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a> in the 1870s; see <a href="/wiki/Icosahedral_symmetry#Related_geometries" title="Icosahedral symmetry">icosahedral symmetry: related geometries</a> for historical discussion and (<a href="#CITEREFKostant1995">Kostant 1995</a>) for more recent exposition. Associated geometries (tilings on <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>) in which the action on <i>p</i> points can be seen are as follows: PSL(2,5) is the symmetries of the icosahedron (genus 0) with the <a href="/wiki/Compound_of_five_tetrahedra" title="Compound of five tetrahedra">compound of five tetrahedra</a> as a 5-element set, PSL(2,7) of the <a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a> (genus 3) with an embedded (complementary) <a href="/wiki/Fano_plane" title="Fano plane">Fano plane</a> as a 7-element set (order 2 biplane), and PSL(2,11) the <b><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509"><span class="vanchor"><span id="buckminsterfullerene_surface"></span><span class="vanchor-text">buckminsterfullerene surface</span></span></b> (genus 70) with embedded <a href="/wiki/Paley_biplane" class="mw-redirect" title="Paley biplane">Paley biplane</a> as an 11-element set (order 3 <a href="/wiki/Biplane_geometry" class="mw-redirect" title="Biplane geometry">biplane</a>).<sup id="cite_ref-martin_16-0" class="reference"><a href="#cite_note-martin-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Of these, the icosahedron dates to antiquity, the Klein quartic to Klein in the 1870s, and the buckyball surface to Pablo Martin and David Singerman in 2008. </p><p>Algebro-geometrically, McKay also associates E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub> respectively with: the <a href="/wiki/27_lines_on_a_cubic_surface" class="mw-redirect" title="27 lines on a cubic surface">27 lines on a cubic surface</a>, the 28 <a href="/wiki/Bitangents_of_a_quartic" title="Bitangents of a quartic">bitangents of a plane quartic curve</a>, and the 120 tritangent planes of a canonic sextic curve of genus 4.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> The first of these is well-known, while the second is connected as follows: projecting the cubic from any point not on a line yields a double cover of the plane, branched along a quartic curve, with the 27 lines mapping to 27 of the 28 bitangents, and the 28th line is the image of the <a href="/wiki/Exceptional_curve" class="mw-redirect" title="Exceptional curve">exceptional curve</a> of the blowup. Note that the <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representations</a> of E<sub>6</sub>, E<sub>7</sub>, E<sub>8</sub> have dimensions 27, 56 (28·2), and 248 (120+128), while the number of roots is 27+45 = 72, 56+70 = 126, and 112+128 = 240. This should also fit into the scheme <sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> of relating E<sub>8,7,6</sub> with the largest three of the sporadic simple groups, Monster, Baby and Fischer 24', cf. <a href="/wiki/Monstrous_moonshine" title="Monstrous moonshine">monstrous moonshine</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Elliptic_surface" title="Elliptic surface">Elliptic surface</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">(<a href="#CITEREFProctor1993">Proctor 1993</a>)</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">(<a href="#CITEREFProctor1993">Proctor 1993</a>, p. 940)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Cameron P.J.; Goethals, J.M.; Seidel, J.J; Shult, E. E. <i>Line graphs, root systems and elliptic geometry</i></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Godsil Chris; Gordon Royle. <i>Algebraic Graph Theory</i>, Chapter 12</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Fedorets A. A., et al. Symmetry of small clusters of levitating water droplets. <i>Phys. Chem. Chem. Phys.</i>, 2020, <a rel="nofollow" class="external free" href="https://doi.org/10.1039/D0CP01804J">https://doi.org/10.1039/D0CP01804J</a></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Arnold, Vladimir, 1997, Toronto Lectures, <i><a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html">Lecture 2: Symplectization, Complexification and Mathematical Trinities</a>,</i> June 1997 (last updated August, 1998). <a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/a2src.zip">TeX</a>, <a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/arnlect2.ps.gz">PostScript</a>, <a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~aar/papers/arnold4.pdf">PDF</a></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/arn-papers.html">Polymathematics: is mathematics a single science or a set of arts?</a></i> On the server since 10-Mar-99, <a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/Polymath.txt">Abstract</a>, <a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/Polymath.tex.gz">TeX</a>, <a rel="nofollow" class="external text" href="http://www.pdmi.ras.ru/~arnsem/Arnold/Polymath.ps.gz">PostScript</a>, <a rel="nofollow" class="external text" href="http://www.neverendingbooks.org/DATA/ArnoldPolymathics.pdf">PDF</a>; see table on page 8</span> </li> <li id="cite_note-arntrin-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-arntrin_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-arntrin_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFle_Bruyn2008" class="citation cs2">le Bruyn, Lieven (17 June 2008), <a rel="nofollow" class="external text" href="http://www.neverendingbooks.org/arnolds-trinities"><i>Arnold's trinities</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arnold%27s+trinities&rft.date=2008-06-17&rft.aulast=le+Bruyn&rft.aufirst=Lieven&rft_id=http%3A%2F%2Fwww.neverendingbooks.org%2Farnolds-trinities&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><i><a rel="nofollow" class="external text" href="http://math.univ-lyon1.fr/~chapoton/trinites.html">Les trinités remarquables</a>,</i> <a rel="nofollow" class="external text" href="http://math.univ-lyon1.fr/~chapoton/">Frédéric Chapoton</a> <span class="languageicon">(in French)</span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFle_Bruyn2008" class="citation cs2">le Bruyn, Lieven (20 June 2008), <a rel="nofollow" class="external text" href="http://www.neverendingbooks.org/arnolds-trinities-version-20"><i>Arnold's trinities version 2.0</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Arnold%27s+trinities+version+2.0&rft.date=2008-06-20&rft.aulast=le+Bruyn&rft.aufirst=Lieven&rft_id=http%3A%2F%2Fwww.neverendingbooks.org%2Farnolds-trinities-version-20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://arxiv4.library.cornell.edu/abs/0810.1465">Arithmetic groups and the affine E<sub>8</sub> Dynkin diagram</a>, by John F. Duncan, in <i>Groups and symmetries: from Neolithic Scots to John McKay</i></span> </li> <li id="cite_note-monster-12"><span class="mw-cite-backlink">^ <a href="#cite_ref-monster_12-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-monster_12-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFle_Bruyn2009" class="citation cs2">le Bruyn, Lieven (22 April 2009), <a rel="nofollow" class="external text" href="http://www.neverendingbooks.org/the-monster-graph-and-mckays-observation"><i>the monster graph and McKay's observation</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=the+monster+graph+and+McKay%27s+observation&rft.date=2009-04-22&rft.aulast=le+Bruyn&rft.aufirst=Lieven&rft_id=http%3A%2F%2Fwww.neverendingbooks.org%2Fthe-monster-graph-and-mckays-observation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKostant1995" class="citation cs2">Kostant, Bertram (1995), <a rel="nofollow" class="external text" href="https://www.ams.org/notices/199509/kostant.pdf">"The Graph of the Truncated Icosahedron and the Last Letter of Galois"</a> <span class="cs1-format">(PDF)</span>, <i>Notices Amer. Math. Soc.</i>, <b>42</b> (4): 959–968, see: The Embedding of PSl(2, 5) into PSl(2, 11) and Galois’ Letter to Chevalier.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Notices+Amer.+Math.+Soc.&rft.atitle=The+Graph+of+the+Truncated+Icosahedron+and+the+Last+Letter+of+Galois&rft.volume=42&rft.issue=4&rft.pages=959-968&rft.date=1995&rft.aulast=Kostant&rft.aufirst=Bertram&rft_id=https%3A%2F%2Fwww.ams.org%2Fnotices%2F199509%2Fkostant.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFle_Bruyn2008" class="citation cs2">le Bruyn, Lieven (12 June 2008), <a rel="nofollow" class="external text" href="http://www.neverendingbooks.org/galois-last-letter"><i>Galois’ last letter</i></a>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100815034546/http://www.neverendingbooks.org/index.php/galois-last-letter.html">archived</a> from the original on 2010-08-15</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois%E2%80%99+last+letter&rft.date=2008-06-12&rft.aulast=le+Bruyn&rft.aufirst=Lieven&rft_id=http%3A%2F%2Fwww.neverendingbooks.org%2Fgalois-last-letter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">(<a href="#CITEREFKostant1995">Kostant 1995</a>, p. 964)</span> </li> <li id="cite_note-martin-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-martin_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMartinSingerman2008" class="citation cs2">Martin, Pablo; Singerman, David (April 17, 2008), <a rel="nofollow" class="external text" href="http://www.neverendingbooks.org/DATA/biplanesingerman.pdf"><i>From Biplanes to the Klein quartic and the Buckyball</i></a> <span class="cs1-format">(PDF)</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=From+Biplanes+to+the+Klein+quartic+and+the+Buckyball&rft.date=2008-04-17&rft.aulast=Martin&rft.aufirst=Pablo&rft.au=Singerman%2C+David&rft_id=http%3A%2F%2Fwww.neverendingbooks.org%2FDATA%2Fbiplanesingerman.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text">Arnold 1997, p. 13</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text">(<a href="#CITEREFMcKaySebbar2007">McKay & Sebbar 2007</a>, p. 11)</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="/wiki/Yang-Hui_He" title="Yang-Hui He">Yang-Hui He</a> and <a href="/wiki/John_McKay_(mathematician)" title="John McKay (mathematician)">John McKay</a>, <a rel="nofollow" class="external free" href="https://arxiv.org/abs/1505.06742">https://arxiv.org/abs/1505.06742</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Sources">Sources</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=9" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1968" class="citation cs2"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, Nicolas</a> (1968), "Chapters 4–6", <i>Groupes et algebres de Lie</i>, Paris: Hermann</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Chapters+4%E2%80%936&rft.btitle=Groupes+et+algebres+de+Lie&rft.place=Paris&rft.pub=Hermann&rft.date=1968&rft.aulast=Bourbaki&rft.aufirst=Nicolas&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArnold1976" class="citation cs2"><a href="/wiki/Vladimir_Arnold" title="Vladimir Arnold">Arnold, Vladimir</a> (1976), "Problems in present day mathematics", in <a href="/wiki/Felix_E._Browder" class="mw-redirect" title="Felix E. Browder">Felix E. Browder</a> (ed.), <i>Mathematical developments arising from Hilbert problems</i>, Proceedings of symposia in pure mathematics, vol. 28, <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=BLnRsA-wRsoC&pg=PA46">46</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Problems+in+present+day+mathematics&rft.btitle=Mathematical+developments+arising+from+Hilbert+problems&rft.series=Proceedings+of+symposia+in+pure+mathematics&rft.pages=46&rft.pub=American+Mathematical+Society&rft.date=1976&rft.aulast=Arnold&rft.aufirst=Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span> Problem VIII. The <i>A-D-E</i> classifications (V. Arnold).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickson1959" class="citation cs2"><a href="/wiki/Leonard_Eugene_Dickson" title="Leonard Eugene Dickson">Dickson, Leonard E.</a> (1959), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=RmlOoBznB8wC&lpg=PA220">"XIII: Groups of the Regular Solids; Quintic Equations"</a>, <i>Algebraic Theories</i>, New York: Dover Publications</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=XIII%3A+Groups+of+the+Regular+Solids%3B+Quintic+Equations&rft.btitle=Algebraic+Theories&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1959&rft.aulast=Dickson&rft.aufirst=Leonard+E.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DRmlOoBznB8wC%26lpg%3DPA220&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkelHesselingSiersmaVeldkamp1977" class="citation cs2"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>; Hesseling; Siersma, JD.; Veldkamp, F. (1977), <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/hazewinkel_et_al.pdf">"The ubiquity of Coxeter Dynkin diagrams. (An introduction of the A-D-E problem)"</a> <span class="cs1-format">(PDF)</span>, <i>Nieuw Archief v. Wiskunde</i>, <b>35</b> (3): 257–307</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nieuw+Archief+v.+Wiskunde&rft.atitle=The+ubiquity+of+Coxeter+Dynkin+diagrams.+%28An+introduction+of+the+A-D-E+problem%29&rft.volume=35&rft.issue=3&rft.pages=257-307&rft.date=1977&rft.aulast=Hazewinkel&rft.aufirst=Michiel&rft.au=Hesseling&rft.au=Siersma%2C+JD.&rft.au=Veldkamp%2C+F.&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2Fhazewinkel_et_al.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKay1980" class="citation cs2"><a href="/wiki/John_McKay_(mathematician)" title="John McKay (mathematician)">McKay, John</a> (1980), "Graphs, singularities and finite groups", <i>Proc. Symp. Pure Math.</i>, <b>37</b>, Amer. Math. Soc.: 183– and 265–</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Symp.+Pure+Math.&rft.atitle=Graphs%2C+singularities+and+finite+groups&rft.volume=37&rft.pages=183-+and+265-&rft.date=1980&rft.aulast=McKay&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKay1982" class="citation cs2"><a href="/wiki/John_McKay_(mathematician)" title="John McKay (mathematician)">McKay, John</a> (1982), "Representations and Coxeter Graphs", <i>"The Geometric Vein", Coxeter Festschrift</i>, Berlin: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, pp. 549–</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Representations+and+Coxeter+Graphs&rft.btitle=%22The+Geometric+Vein%22%2C+Coxeter+Festschrift&rft.place=Berlin&rft.pages=549-&rft.pub=Springer-Verlag&rft.date=1982&rft.aulast=McKay&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKac1990" class="citation cs2">Kac, Victor G. (1990), <i>Infinite-Dimensional Lie Algebras</i> (3rd ed.), Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-46693-8" title="Special:BookSources/0-521-46693-8"><bdi>0-521-46693-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Infinite-Dimensional+Lie+Algebras&rft.place=Cambridge&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=1990&rft.isbn=0-521-46693-8&rft.aulast=Kac&rft.aufirst=Victor+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKay2001" class="citation cs2"><a href="/wiki/John_McKay_(mathematician)" title="John McKay (mathematician)">McKay, John</a> (January 1, 2001), <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/ADE.html"><i>A Rapid Introduction to ADE Theory</i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Rapid+Introduction+to+ADE+Theory&rft.date=2001-01-01&rft.aulast=McKay&rft.aufirst=John&rft_id=http%3A%2F%2Fmath.ucr.edu%2Fhome%2Fbaez%2FADE.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFProctor1993" class="citation cs2">Proctor, R. A. (December 1993), "Two Amusing Dynkin Diagram Graph Classifications", <i><a href="/wiki/The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a></i>, <b>100</b> (10): 937–941, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2324217">10.2307/2324217</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2324217">2324217</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Two+Amusing+Dynkin+Diagram+Graph+Classifications&rft.volume=100&rft.issue=10&rft.pages=937-941&rft.date=1993-12&rft.issn=0002-9890&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2324217%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F2324217&rft.aulast=Proctor&rft.aufirst=R.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKaySebbar2007" class="citation book cs1">McKay, J.; Sebbar, Abdellah (2007). "Replicable Functions: An introduction". <i>Frontiers in Number Theory, Physics, and Geometry, II</i>. Springer. pp. 373–386. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-30308-4_10">10.1007/978-3-540-30308-4_10</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Replicable+Functions%3A+An+introduction&rft.btitle=Frontiers+in+Number+Theory%2C+Physics%2C+and+Geometry%2C+II&rft.pages=373-386&rft.pub=Springer&rft.date=2007&rft_id=info%3Adoi%2F10.1007%2F978-3-540-30308-4_10&rft.aulast=McKay&rft.aufirst=J.&rft.au=Sebbar%2C+Abdellah&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStekolshchik2008" class="citation cs2">Stekolshchik, R. (2008), <i>Notes on Coxeter Transformations and the McKay Correspondence</i>, Springer Monographs in Mathematics, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-540-77398-3">10.1007/978-3-540-77398-3</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-77398-6" title="Special:BookSources/978-3-540-77398-6"><bdi>978-3-540-77398-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Notes+on+Coxeter+Transformations+and+the+McKay+Correspondence&rft.series=Springer+Monographs+in+Mathematics&rft.date=2008&rft_id=info%3Adoi%2F10.1007%2F978-3-540-77398-3&rft.isbn=978-3-540-77398-6&rft.aulast=Stekolshchik&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_Hoboken2002" class="citation cs2">van Hoboken, Joris (2002), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120426001310/http://www.jorisvanhoboken.nl/wp-content/uploads/2007/03/platonic-solids-binary-polyhedral-groups-kleinian-singularities-and-lie-algebras-of-type-ade.pdf"><i>Platonic solids, binary polyhedral groups, Kleinian singularities and Lie algebras of type A,D,E</i></a> <span class="cs1-format">(PDF)</span>, Master's Thesis, University of Amsterdam, archived from <a rel="nofollow" class="external text" href="http://www.jorisvanhoboken.nl/wp-content/uploads/2007/03/platonic-solids-binary-polyhedral-groups-kleinian-singularities-and-lie-algebras-of-type-ade.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2012-04-26<span class="reference-accessdate">, retrieved <span class="nowrap">2011-11-23</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Platonic+solids%2C+binary+polyhedral+groups%2C+Kleinian+singularities+and+Lie+algebras+of+type+A%2CD%2CE&rft.series=Master%27s+Thesis&rft.pub=University+of+Amsterdam&rft.date=2002&rft.aulast=van+Hoboken&rft.aufirst=Joris&rft_id=http%3A%2F%2Fwww.jorisvanhoboken.nl%2Fwp-content%2Fuploads%2F2007%2F03%2Fplatonic-solids-binary-polyhedral-groups-kleinian-singularities-and-lie-algebras-of-type-ade.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AADE+classification" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=ADE_classification&action=edit&section=10" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/John_C._Baez" title="John C. Baez">John Baez</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/TWF.html">This Week's Finds in Mathematical Physics</a>: <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week62.html">Week 62</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week63.html">Week 63</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week64.html">Week 64</a>, <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week65.html">Week 65</a>, August 28, 1995, through October 3, 1995, and <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/week230.html">Week 230</a>, May 4, 2006</li> <li><a rel="nofollow" class="external text" href="http://www.valdostamuseum.com/hamsmith/McKay.html">The McKay Correspondence</a>, Tony Smith</li> <li><a rel="nofollow" class="external text" href="http://motls.blogspot.com/2006/05/ade-classification-mckay.html">ADE classification, McKay correspondence, and string theory</a>, <a href="/wiki/Lubo%C5%A1_Motl" title="Luboš Motl">Luboš Motl</a>, <i><a rel="nofollow" class="external text" href="http://motls.blogspot.com/">The Reference Frame</a>,</i> May 7, 2006</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output 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