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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Real and complex forms</span> </div> </a> <ul id="toc-Real_and_complex_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-E8_as_an_algebraic_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#E8_as_an_algebraic_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>E<sub>8</sub> as an algebraic group</span> </div> </a> <ul id="toc-E8_as_an_algebraic_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Constructions</span> </div> </a> <ul id="toc-Constructions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geometry" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Geometry"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Geometry</span> </div> </a> <ul id="toc-Geometry-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-E8_root_system" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#E8_root_system"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>E<sub>8</sub> root system</span> </div> </a> <button aria-controls="toc-E8_root_system-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle E<sub>8</sub> root system subsection</span> </button> <ul id="toc-E8_root_system-sublist" class="vector-toc-list"> <li id="toc-Construction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Construction</span> </div> </a> <ul id="toc-Construction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynkin_diagram" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynkin_diagram"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Dynkin diagram</span> </div> </a> <ul id="toc-Dynkin_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cartan_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cartan_matrix"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Cartan matrix</span> </div> </a> <ul id="toc-Cartan_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Simple roots</span> </div> </a> <ul id="toc-Simple_roots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weyl_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weyl_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.5</span> <span>Weyl group</span> </div> </a> <ul id="toc-Weyl_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-E8_root_lattice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#E8_root_lattice"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.6</span> <span>E<sub>8</sub> root lattice</span> </div> </a> <ul id="toc-E8_root_lattice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_subalgebras_of_E8" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_subalgebras_of_E8"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.7</span> <span>Simple subalgebras of E<sub>8</sub></span> </div> </a> <ul id="toc-Simple_subalgebras_of_E8-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Chevalley_groups_of_type_E8" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Chevalley_groups_of_type_E8"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Chevalley groups of type E<sub>8</sub></span> </div> </a> <ul id="toc-Chevalley_groups_of_type_E8-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subgroups,_subalgebras,_and_extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Subgroups,_subalgebras,_and_extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Subgroups, subalgebras, and extensions</span> </div> </a> <ul id="toc-Subgroups,_subalgebras,_and_extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>History</span> </div> </a> <ul id="toc-History-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">11</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 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</div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">E<sub>8</sub> (mathematics)</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/E8_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="E8 (математика) – Chuvash" lang="cv" hreflang="cv" data-title="E8 (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/E8_(matematik)" title="E8 (matematik) – Danish" lang="da" hreflang="da" data-title="E8 (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/E8_(Gruppe)" title="E8 (Gruppe) – German" lang="de" hreflang="de" data-title="E8 (Gruppe)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/E8_(matem%C3%A1ticas)" title="E8 (matemáticas) – Spanish" lang="es" hreflang="es" data-title="E8 (matemáticas)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/E8_(math%C3%A9matiques)" title="E8 (mathématiques) – French" lang="fr" hreflang="fr" data-title="E8 (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/E%E2%82%88" title="E₈ – Korean" lang="ko" hreflang="ko" data-title="E₈" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/E8_(matematica)" title="E8 (matematica) – Italian" lang="it" hreflang="it" data-title="E8 (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/E8_(wiskunde)" title="E8 (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="E8 (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/E8_(%E6%95%B0%E5%AD%A6)" title="E8 (数学) – Japanese" lang="ja" hreflang="ja" data-title="E8 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><table class="sidebar sidebar-collapse nomobile nowraplinks" style="width:20.0em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → <b>Group theory</b></span><br /><a href="/wiki/Group_theory" title="Group theory">Group theory</a></th></tr><tr><td class="sidebar-image"><span class="skin-invert"><span typeof="mw:File"><a href="/wiki/File:Cyclic_group.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/120px-Cyclic_group.svg.png" decoding="async" width="120" height="117" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/180px-Cyclic_group.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Cyclic_group.svg/240px-Cyclic_group.svg.png 2x" data-file-width="443" data-file-height="431" /></a></span></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Basic notions</div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Subgroup" title="Subgroup">Subgroup</a></li> <li><a href="/wiki/Normal_subgroup" title="Normal subgroup">Normal subgroup</a></li> <li><a href="/wiki/Group_action" title="Group action">Group action</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Quotient_group" title="Quotient group">Quotient group</a></li> <li><a href="/wiki/Semidirect_product" title="Semidirect product">(Semi-)</a><a href="/wiki/Direct_product_of_groups" title="Direct product of groups">direct product</a></li> <li><a href="/wiki/Direct_sum_of_groups" title="Direct sum of groups">Direct sum</a></li> <li><a href="/wiki/Free_product" title="Free product">Free product</a></li> <li><a href="/wiki/Wreath_product" title="Wreath product">Wreath product</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <i><a href="/wiki/Group_homomorphism" title="Group homomorphism">Group homomorphisms</a></i></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Kernel_(algebra)#Group_homomorphisms" title="Kernel (algebra)">kernel</a></li> <li><a href="/wiki/Image_(mathematics)" title="Image (mathematics)">image</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_group" title="Simple group">simple</a></li> <li><a href="/wiki/Finite_group" title="Finite group">finite</a></li> <li><a href="/wiki/Infinite_group" title="Infinite group">infinite</a></li> <li><a href="/wiki/Continuous_group" class="mw-redirect" title="Continuous group">continuous</a></li> <li><a href="/wiki/Multiplicative_group" title="Multiplicative group">multiplicative</a></li> <li><a href="/wiki/Additive_group" title="Additive group">additive</a></li> <li><a href="/wiki/Cyclic_group" title="Cyclic group">cyclic</a></li> <li><a href="/wiki/Abelian_group" title="Abelian group">abelian</a></li> <li><a href="/wiki/Dihedral_group" title="Dihedral group">dihedral</a></li> <li><a href="/wiki/Nilpotent_group" title="Nilpotent group">nilpotent</a></li> <li><a href="/wiki/Solvable_group" title="Solvable group">solvable</a></li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Glossary_of_group_theory" title="Glossary of group theory">Glossary of group theory</a></li> <li><a href="/wiki/List_of_group_theory_topics" title="List of group theory topics">List of group theory topics</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Finite_group" title="Finite group">Finite groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cyclic_group" title="Cyclic group">Cyclic group</a> Z<sub><i>n</i></sub></li> <li><a href="/wiki/Symmetric_group" title="Symmetric group">Symmetric group</a> S<sub><i>n</i></sub></li> <li><a href="/wiki/Alternating_group" title="Alternating group">Alternating group</a> A<sub><i>n</i></sub></li></ul> <ul><li><a href="/wiki/Dihedral_group" title="Dihedral group">Dihedral group</a> D<sub><i>n</i></sub></li> <li><a href="/wiki/Quaternion_group" title="Quaternion group">Quaternion group</a> Q</li></ul></td> </tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Cauchy%27s_theorem_(group_theory)" title="Cauchy's theorem (group theory)">Cauchy's theorem</a></li> <li><a href="/wiki/Lagrange%27s_theorem_(group_theory)" title="Lagrange's theorem (group theory)">Lagrange's theorem</a></li></ul> <ul><li><a href="/wiki/Sylow_theorems" title="Sylow theorems">Sylow theorems</a></li> <li><a href="/wiki/Hall_subgroup" title="Hall subgroup">Hall's theorem</a></li></ul> <ul><li><a href="/wiki/P-group" title="P-group"><i>p</i>-group</a></li> <li><a href="/wiki/Elementary_abelian_group" title="Elementary abelian group">Elementary abelian group</a></li></ul> <ul><li><a href="/wiki/Frobenius_group" title="Frobenius group">Frobenius group</a></li></ul> <ul><li><a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a></li></ul></td> </tr><tr><th class="sidebar-heading"> <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">Classification of finite simple groups</a></th></tr><tr><td class="sidebar-content"> <ul><li>cyclic</li> <li>alternating</li> <li><a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Lie type</a></li> <li><a href="/wiki/Sporadic_group" title="Sporadic group">sporadic</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li><a href="/wiki/Discrete_group" title="Discrete group">Discrete groups</a></li><li><a href="/wiki/Lattice_(discrete_subgroup)" title="Lattice (discrete subgroup)">Lattices</a></li></ul></div></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Integer" title="Integer">Integers</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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li> <li><a href="/wiki/Free_group" title="Free group">Free group</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Modular_group" title="Modular group">Modular groups</a> <div class="hlist"><ul><li>PSL(2, <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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li><li>SL(2, <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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>)</li></ul></div></div> <ul><li><a href="/wiki/Arithmetic_group" title="Arithmetic group">Arithmetic group</a></li> <li><a href="/wiki/Lattice_(group)" title="Lattice (group)">Lattice</a></li> <li><a href="/wiki/Hyperbolic_group" title="Hyperbolic group">Hyperbolic group</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Topological_group" title="Topological group">Topological</a> and <a href="/wiki/Lie_group" title="Lie group">Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Solenoid_(mathematics)" title="Solenoid (mathematics)">Solenoid</a></li> <li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li></ul> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a> E(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li></ul> <ul><li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a class="mw-selflink selflink">E<sub>8</sub></a></li></ul> <ul><li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal</a></li></ul> <ul><li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li></ul> <div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><a href="/wiki/Infinite_dimensional_Lie_group" class="mw-redirect" title="Infinite dimensional Lie group">Infinite dimensional Lie group</a> <div class="hlist"><ul><li>O(∞)</li><li>SU(∞)</li><li>Sp(∞)</li></ul></div></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Algebraic_group" title="Algebraic group">Algebraic groups</a></div><div class="sidebar-list-content mw-collapsible-content hlist" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">Linear algebraic group</a></li></ul> <ul><li><a href="/wiki/Reductive_group" title="Reductive group">Reductive group</a></li></ul> <ul><li><a href="/wiki/Abelian_variety" title="Abelian variety">Abelian variety</a></li></ul> <ul><li><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curve</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="/wiki/Lie_group" title="Lie group">Lie groups</a> and <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></th></tr><tr><td class="sidebar-image" style="padding-bottom:0.9em;"><span typeof="mw:File/Frameless"><a href="/wiki/File:E8Petrie.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/180px-E8Petrie.svg.png" decoding="async" width="180" height="181" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/270px-E8Petrie.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/E8Petrie.svg/360px-E8Petrie.svg.png 2x" data-file-width="2852" data-file-height="2863" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Classical_group" title="Classical group">Classical groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/General_linear_group" title="General linear group">General linear</a> GL(<i>n</i>)</li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear</a> SL(<i>n</i>)</li> <li><a href="/wiki/Orthogonal_group" title="Orthogonal group">Orthogonal</a> O(<i>n</i>)</li> <li><a href="/wiki/Special_orthogonal_group" class="mw-redirect" title="Special orthogonal group">Special orthogonal</a> SO(<i>n</i>)</li> <li><a href="/wiki/Unitary_group" title="Unitary group">Unitary</a> U(<i>n</i>)</li> <li><a href="/wiki/Special_unitary_group" title="Special unitary group">Special unitary</a> SU(<i>n</i>)</li> <li><a href="/wiki/Symplectic_group" title="Symplectic group">Symplectic</a> Sp(<i>n</i>)</li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Simple_Lie_group" title="Simple Lie group">Simple Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks hlist" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Classical</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Simple_Lie_group#A_series" title="Simple Lie group">A<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#B_series" title="Simple Lie group">B<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#C_series" title="Simple Lie group">C<sub><i>n</i></sub></a></li> <li><a href="/wiki/Simple_Lie_group#D_series" title="Simple Lie group">D<sub><i>n</i></sub></a></li></ul></td> </tr><tr><th class="sidebar-heading" style="font-weight:normal; font-style:italic;"> Exceptional</th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a></li> <li><a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a></li> <li><a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a></li> <li><a class="mw-selflink selflink">E<sub>8</sub></a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Other Lie groups</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Circle_group" title="Circle group">Circle</a></li> <li><a href="/wiki/Lorentz_group" title="Lorentz group">Lorentz</a></li> <li><a href="/wiki/Poincar%C3%A9_group" title="Poincaré group">Poincaré</a></li> <li><a href="/wiki/Conformal_group" title="Conformal group">Conformal group</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a></li> <li><a href="/wiki/Loop_group" title="Loop group">Loop</a></li> <li><a href="/wiki/Euclidean_group" title="Euclidean group">Euclidean</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie group–Lie algebra correspondence</a></li> <li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">Exponential map</a></li> <li><a href="/wiki/Adjoint_representation" title="Adjoint representation">Adjoint representation</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Killing_form" title="Killing form">Killing form</a></li><li><a href="/wiki/Index_of_a_Lie_algebra" title="Index of a Lie algebra">Index</a></li></ul></div></li> <li><a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">Simple Lie algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Affine_Lie_algebra" title="Affine Lie algebra">Affine Lie algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Semisimple_Lie_algebra" title="Semisimple Lie algebra">Semisimple Lie algebra</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagrams</a></li> <li><a href="/wiki/Cartan_subalgebra" title="Cartan subalgebra">Cartan subalgebra</a></li> <li><div class="hlist"><ul><li><a href="/wiki/Root_system" title="Root system">Root system</a></li><li><a href="/wiki/Weyl_group" title="Weyl group">Weyl group</a></li></ul></div></li> <li><div class="hlist"><ul><li><a href="/wiki/Real_form_(Lie_theory)" title="Real form (Lie theory)">Real form</a></li><li><a href="/wiki/Complexification_(Lie_group)" title="Complexification (Lie group)">Complexification</a></li></ul></div></li> <li><a href="/wiki/Split_Lie_algebra" title="Split Lie algebra">Split Lie algebra</a></li> <li><a href="/wiki/Compact_Lie_algebra" title="Compact Lie algebra">Compact Lie algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Representation_theory" title="Representation theory">Representation theory</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Representation_of_a_Lie_group" title="Representation of a Lie group">Lie group representation</a></li> <li><a href="/wiki/Lie_algebra_representation" title="Lie algebra representation">Lie algebra representation</a></li> <li><a href="/wiki/Representation_theory_of_semisimple_Lie_algebras" title="Representation theory of semisimple Lie algebras">Representation theory of semisimple Lie algebras</a></li> <li><a href="/wiki/Representations_of_classical_Lie_groups" title="Representations of classical Lie groups">Representations of classical Lie groups</a></li> <li><a href="/wiki/Theorem_of_the_highest_weight" title="Theorem of the highest weight">Theorem of the highest weight</a></li> <li><a href="/wiki/Borel%E2%80%93Weil%E2%80%93Bott_theorem" title="Borel–Weil–Bott theorem">Borel–Weil–Bott theorem</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Lie groups in <a href="/wiki/Physics" title="Physics">physics</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="plainlist"> <ul><li><a href="/wiki/Particle_physics_and_representation_theory" title="Particle physics and representation theory">Particle physics and representation theory</a></li> <li><a href="/wiki/Representation_theory_of_the_Lorentz_group" title="Representation theory of the Lorentz group">Lorentz group representations</a></li> <li><a href="/wiki/Representation_theory_of_the_Poincar%C3%A9_group" title="Representation theory of the Poincaré group">Poincaré group representations</a></li> <li><a href="/wiki/Representation_theory_of_the_Galilean_group" title="Representation theory of the Galilean group">Galilean group representations</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Sophus_Lie" title="Sophus Lie">Sophus Lie</a></li> <li><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></li> <li><a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a></li> <li><a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li> <li><a href="/wiki/Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li> <li><a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a></li> <li><a href="/wiki/Harish-Chandra" title="Harish-Chandra">Harish-Chandra</a></li> <li><a href="/wiki/Armand_Borel" title="Armand Borel">Armand Borel</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-below plainlist"> <ul><li><a href="/wiki/Glossary_of_Lie_groups_and_Lie_algebras" title="Glossary of Lie groups and Lie algebras">Glossary</a></li> <li><a href="/wiki/Table_of_Lie_groups" title="Table of Lie groups">Table of Lie groups</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Lie_groups" title="Template:Lie groups"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Lie_groups" title="Template talk:Lie groups"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Lie_groups" title="Special:EditPage/Template:Lie groups"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>E<sub>8</sub></b> is any of several closely related <a href="/wiki/Exceptional_simple_Lie_group" class="mw-redirect" title="Exceptional simple Lie group">exceptional simple Lie groups</a>, linear <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a> or Lie algebras of <a href="/wiki/Dimension" title="Dimension">dimension</a> 248; the same notation is used for the corresponding <a href="/wiki/Root_lattice" class="mw-redirect" title="Root lattice">root lattice</a>, which has <a href="/wiki/Rank_of_a_Lie_group" class="mw-redirect" title="Rank of a Lie group">rank</a> 8. The designation E<sub>8</sub> comes from the <a href="/wiki/Killing_form" title="Killing form">Cartan–Killing classification</a> of the complex <a href="/wiki/Simple_Lie_algebra" title="Simple Lie algebra">simple Lie algebras</a>, which fall into four infinite series labeled A<sub><i>n</i></sub>, B<sub><i>n</i></sub>, C<sub><i>n</i></sub>, D<sub><i>n</i></sub>, and <a href="/wiki/Exceptional_simple_Lie_group" class="mw-redirect" title="Exceptional simple Lie group">five exceptional cases</a> labeled <a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, and E<sub>8</sub>. The E<sub>8</sub> algebra is the largest and most complicated of these exceptional cases. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Basic_description">Basic description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=1" title="Edit section: Basic description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Lie_group" title="Lie group">Lie group</a> E<sub>8</sub> has dimension 248. Its <a href="/wiki/Cartan_subgroup" title="Cartan subgroup">rank</a>, which is the dimension of its <a href="/wiki/Maximal_torus" title="Maximal torus">maximal torus</a>, is eight. </p><p>Therefore, the vectors of the root system are in eight-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>: they are described explicitly later in this article. The <a href="/wiki/Weyl_group" title="Weyl group">Weyl group</a> of E<sub>8</sub>, which is the <a href="/wiki/Symmetry_group" title="Symmetry group">group of symmetries</a> of the maximal torus that are induced by <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugations</a> in the whole group, has order 2<sup>14</sup> 3<sup>5</sup> 5<sup>2</sup> 7 = <span style="white-space:nowrap">696<span style="margin-left:0.25em">729</span><span style="margin-left:0.25em">600</span></span>. </p><p>The compact group E<sub>8</sub> is unique among simple compact Lie groups in that its non-<a href="/wiki/Trivial_(mathematics)" class="mw-redirect" title="Trivial (mathematics)">trivial</a> representation of smallest dimension is the <a href="/wiki/Adjoint_representation_of_a_Lie_algebra" class="mw-redirect" title="Adjoint representation of a Lie algebra">adjoint representation</a> (of dimension 248) acting on the Lie algebra E<sub>8</sub> itself; it is also the unique one that has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length). </p><p>There is a Lie algebra <a href="/wiki/En_(Lie_algebra)" title="En (Lie algebra)">E<sub><i>k</i></sub></a> for every integer <i>k</i> ≥ 3. The largest value of <i>k</i> for which E<sub><i>k</i></sub> is finite-dimensional is <i>k</i> = 8, that is, E<sub><i>k</i></sub> is infinite-dimensional for any <i>k</i> > 8. </p> <div class="mw-heading mw-heading2"><h2 id="Real_and_complex_forms">Real and complex forms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=2" title="Edit section: Real and complex forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There is a unique complex Lie algebra of type E<sub>8</sub>, corresponding to a complex group of complex dimension 248. The complex Lie group E<sub>8</sub> of <a href="/wiki/Complex_dimension" title="Complex dimension">complex dimension</a> 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal <a href="/wiki/Compact_space" title="Compact space">compact</a> subgroup the compact form (see below) of E<sub>8</sub>, and has an outer automorphism group of order 2 generated by complex conjugation. </p><p>As well as the complex Lie group of type E<sub>8</sub>, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows: </p> <ul><li>The compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.</li> <li>The split form, EVIII (or E<sub>8(8)</sub>), which has maximal compact subgroup Spin(16)/(<b>Z</b>/2<b>Z</b>), fundamental group of order 2 (implying that it has a <a href="/wiki/Double_covering_group" class="mw-redirect" title="Double covering group">double cover</a>, which is a simply connected Lie real group but is not algebraic, see <a href="#E8_as_an_algebraic_group">below</a>) and has trivial outer automorphism group.</li> <li>EIX (or E<sub>8(−24)</sub>), which has maximal compact subgroup E<sub>7</sub>×SU(2)/(−1,−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) and has trivial outer automorphism group.</li></ul> <p>For a complete list of real forms of simple Lie algebras, see the <a href="/wiki/List_of_simple_Lie_groups" class="mw-redirect" title="List of simple Lie groups">list of simple Lie groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="E8_as_an_algebraic_group">E<sub>8</sub> as an algebraic group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=3" title="Edit section: E8 as an algebraic group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By means of a <a href="/wiki/Chevalley_basis" title="Chevalley basis">Chevalley basis</a> for the Lie algebra, one can define E<sub>8</sub> as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") form of E<sub>8</sub>. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or "twists" of E<sub>8</sub>, which are classified in the general framework of <a href="/wiki/Galois_cohomology" title="Galois cohomology">Galois cohomology</a> (over a <a href="/wiki/Perfect_field" title="Perfect field">perfect field</a> <i>k</i>) by the set H<sup>1</sup>(<i>k</i>,Aut(E<sub>8</sub>)), which, because the Dynkin diagram of E<sub>8</sub> (see <a href="#Dynkin_diagram">below</a>) has no automorphisms, coincides with H<sup>1</sup>(<i>k</i>,E<sub>8</sub>).<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> </p><p>Over <b>R</b>, the real connected component of the identity of these algebraically twisted forms of E<sub>8</sub> coincide with the three real Lie groups mentioned <a href="#Real_and_complex_forms">above</a>, but with a subtlety concerning the fundamental group: all forms of E<sub>8</sub> are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E<sub>8</sub> are therefore not algebraic and admit no faithful finite-dimensional representations. </p><p>Over finite fields, the <a href="/wiki/Lang%E2%80%93Steinberg_theorem" class="mw-redirect" title="Lang–Steinberg theorem">Lang–Steinberg theorem</a> implies that H<sup>1</sup>(<i>k</i>,E<sub>8</sub>)=0, meaning that E<sub>8</sub> has no twisted forms: see <a href="#Chevalley_groups_of_type_E8">below</a>. </p><p>The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the <a href="/wiki/Weyl_character_formula" title="Weyl character formula">Weyl character formula</a>. The dimensions of the smallest irreducible representations are (sequence <span class="nowrap external"><a href="//oeis.org/A121732" class="extiw" title="oeis:A121732">A121732</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>): </p> <dl><dd>1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000 (twice), 12692520960...</dd></dl> <p>The 248-dimensional representation is the <a href="/wiki/Adjoint_representation_of_a_Lie_group" class="mw-redirect" title="Adjoint representation of a Lie group">adjoint representation</a>. There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 (sequence <span class="nowrap external"><a href="//oeis.org/A181746" class="extiw" title="oeis:A181746">A181746</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)). The <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representations</a> are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the <a href="#Dynkin_diagram">Dynkin diagram</a> in the order chosen for the <a href="#Cartan_matrix">Cartan matrix</a> below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third). </p><p>The coefficients of the character formulas for infinite dimensional irreducible <a href="/wiki/Representation_theory" title="Representation theory">representations</a> of E<sub>8</sub> depend on some large square matrices consisting of polynomials, the <a href="/wiki/Lusztig%E2%80%93Vogan_polynomial" class="mw-redirect" title="Lusztig–Vogan polynomial">Lusztig–Vogan polynomials</a>, an analogue of <a href="/wiki/Kazhdan%E2%80%93Lusztig_polynomial" title="Kazhdan–Lusztig polynomial">Kazhdan–Lusztig polynomials</a> introduced for <a href="/wiki/Reductive_group" title="Reductive group">reductive groups</a> in general by <a href="/wiki/George_Lusztig" title="George Lusztig">George Lusztig</a> and <a href="/wiki/David_Kazhdan" title="David Kazhdan">David Kazhdan</a> (1983). The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations. </p><p>These matrices were computed after four years of collaboration by a <a href="/wiki/Atlas_of_Lie_groups_and_representations" title="Atlas of Lie groups and representations">group of 18 mathematicians and computer scientists</a>, led by <a href="/wiki/Jeffrey_Adams_(mathematician)" title="Jeffrey Adams (mathematician)">Jeffrey Adams</a>, with much of the programming done by <a href="/wiki/Fokko_du_Cloux" title="Fokko du Cloux">Fokko du Cloux</a>. The most difficult case (for exceptional groups) is the split <a href="/wiki/Real_form" class="mw-redirect" title="Real form">real form</a> of E<sub>8</sub> (see above), where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of <i>E</i><sub>8</sub> is far longer than any other case. The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it. </p><p>The representations of the E<sub>8</sub> groups over finite fields are given by <a href="/wiki/Deligne%E2%80%93Lusztig_theory" title="Deligne–Lusztig theory">Deligne–Lusztig theory</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=4" title="Edit section: Constructions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can construct the (compact form of the) E<sub>8</sub> group as the <a href="/wiki/Automorphism_group" title="Automorphism group">automorphism group</a> of the corresponding <b>e</b><sub>8</sub> Lie algebra. This algebra has a 120-dimensional subalgebra <b>so</b>(16) generated by <i>J</i><sub><i>ij</i></sub> as well as 128 new generators <i>Q</i><sub><i>a</i></sub> that transform as a <a href="/wiki/Weyl%E2%80%93Majorana_spinor" class="mw-redirect" title="Weyl–Majorana spinor">Weyl–Majorana spinor</a> of <b>spin</b>(16). These statements determine the commutators </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 \left[J_{ij},J_{k\ell }\right]=\delta _{jk}J_{i\ell }-\delta _{j\ell }J_{ik}-\delta _{ik}J_{j\ell }+\delta _{i\ell }J_{jk}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>ℓ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ℓ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>ℓ</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[J_{ij},J_{k\ell }\right]=\delta _{jk}J_{i\ell }-\delta _{j\ell }J_{ik}-\delta _{ik}J_{j\ell }+\delta _{i\ell }J_{jk}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bec09a9b82bdba74164819ce65265a0a00b09979" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:42.076ex; height:3.009ex;" alt="{\displaystyle \left[J_{ij},J_{k\ell }\right]=\delta _{jk}J_{i\ell }-\delta _{j\ell }J_{ik}-\delta _{ik}J_{j\ell }+\delta _{i\ell }J_{jk}}"></span></dd></dl> <p>as well as </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 \left[J_{ij},Q_{a}\right]={\frac {1}{4}}\left(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i}\right)_{ab}Q_{b},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>b</mi> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[J_{ij},Q_{a}\right]={\frac {1}{4}}\left(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i}\right)_{ab}Q_{b},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c26842f45b69aa8321bb2824eb3cae60a765fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.248ex; height:5.176ex;" alt="{\displaystyle \left[J_{ij},Q_{a}\right]={\frac {1}{4}}\left(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i}\right)_{ab}Q_{b},}"></span></dd></dl> <p>while the remaining commutators (not anticommutators!) between the spinor generators are defined as </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 \left[Q_{a},Q_{b}\right]=\gamma _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mi>i</mi> </mrow> </msubsup> <msubsup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>b</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo stretchy="false">]</mo> </mrow> </msubsup> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[Q_{a},Q_{b}\right]=\gamma _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4d9a1b2b940688a05ba4c1fe3c678cf45c498ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.428ex; height:3.676ex;" alt="{\displaystyle \left[Q_{a},Q_{b}\right]=\gamma _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.}"></span></dd></dl> <p>It is then possible to check that the <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a> is satisfied. </p> <div class="mw-heading mw-heading2"><h2 id="Geometry">Geometry</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=5" title="Edit section: Geometry"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The compact real form of E<sub>8</sub> is the <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a> of the 128-dimensional exceptional compact <a href="/wiki/Riemannian_symmetric_space" class="mw-redirect" title="Riemannian symmetric space">Riemannian symmetric space</a> EVIII (in Cartan's <a href="/wiki/Riemannian_symmetric_space#Classification_of_Riemannian_symmetric_spaces" class="mw-redirect" title="Riemannian symmetric space">classification</a>). It is known informally as the "<a href="/wiki/Octooctonionic_projective_plane" class="mw-redirect" title="Octooctonionic projective plane">octooctonionic projective plane</a>" because it can be built using an algebra that is the tensor product of the <a href="/wiki/Octonion" title="Octonion">octonions</a> with themselves, and is also known as a <a href="/wiki/Rosenfeld_projective_plane" class="mw-redirect" title="Rosenfeld projective plane">Rosenfeld projective plane</a>, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the <a href="/wiki/Freudenthal_magic_square" title="Freudenthal magic square"><i>magic square</i></a>, due to <a href="/wiki/Hans_Freudenthal" title="Hans Freudenthal">Hans Freudenthal</a> and <a href="/wiki/Jacques_Tits" title="Jacques Tits">Jacques Tits</a> (<a href="#CITEREFLandsbergManivel2001">Landsberg & Manivel 2001</a>). </p> <div class="mw-heading mw-heading2"><h2 id="E8_root_system">E<sub>8</sub> root system</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=6" title="Edit section: E8 root system"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:E8_roots_zome.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/2/2b/E8_roots_zome.jpg/220px-E8_roots_zome.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/2/2b/E8_roots_zome.jpg/330px-E8_roots_zome.jpg 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/2/2b/E8_roots_zome.jpg/440px-E8_roots_zome.jpg 2x" data-file-width="768" data-file-height="768" /></a><figcaption><a href="/wiki/Zome" class="mw-redirect" title="Zome">Zome</a> model of the E<sub>8</sub> root system, projected into three-space, and represented by the vertices of the <a href="/wiki/4_21_polytope" title="4 21 polytope">4<sub>21</sub> polytope</a>, <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c7/CDel_nodea.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/56/CDel_3a.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c7/CDel_nodea.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/56/CDel_3a.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/4/43/CDel_branch.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/56/CDel_3a.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c7/CDel_nodea.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/56/CDel_3a.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c7/CDel_nodea.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/56/CDel_3a.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c7/CDel_nodea.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/56/CDel_3a.png" decoding="async" width="6" height="23" class="mw-file-element" data-file-width="6" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/a/aa/CDel_nodea_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:4_21_E8_to_3D_H3_symmetry_concentric_hulls.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/4_21_E8_to_3D_H3_symmetry_concentric_hulls.png/220px-4_21_E8_to_3D_H3_symmetry_concentric_hulls.png" decoding="async" width="220" height="316" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/33/4_21_E8_to_3D_H3_symmetry_concentric_hulls.png/330px-4_21_E8_to_3D_H3_symmetry_concentric_hulls.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/33/4_21_E8_to_3D_H3_symmetry_concentric_hulls.png/440px-4_21_E8_to_3D_H3_symmetry_concentric_hulls.png 2x" data-file-width="594" data-file-height="853" /></a><figcaption>Shown in 3D projection using the basis vectors [<i>u</i>,<i>v</i>,<i>w</i>] giving H3 symmetry: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"><ul><li><i>u</i> = (1, <i>φ</i>, 0, −1, <i>φ</i>, 0,0,0)</li><li><i>v</i> = (<i>φ</i>, 0, 1, <i>φ</i>, 0, −1,0,0)</li><li><i>w</i> = (0, 1, <i>φ</i>, 0, −1, <i>φ</i>,0,0)</li></ul></div> The projected <a href="/wiki/4_21_polytope" title="4 21 polytope">4<sub>21</sub> polytope</a> vertices are sorted and tallied by their 3D norm generating the increasingly transparent hulls of each set of tallied norms. These show: <div><ol style="list-style-position:inside; margin:0;"><li>4 points at the origin</li><li>2 icosahedrons</li><li>2 dodecahedrons</li><li>4 icosahedrons</li><li>1 icosidodecahedron</li><li>2 dodecahedrons</li><li>2 icosahedrons</li><li>1 icosidodecahedron</li></ol></div> for 240 vertices. These are two concentric sets of hulls from the H4 symmetry of the <a href="/wiki/600-cell" title="600-cell">600-cell</a> scaled by the golden ratio.<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> </figcaption></figure> <p>A <a href="/wiki/Root_system" title="Root system">root system</a> of rank <i>r</i> is a particular finite configuration of vectors, called <i>roots</i>, which span an <i>r</i>-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> and satisfy certain geometrical properties. In particular, the root system must be invariant under <a href="/wiki/Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a> through the hyperplane perpendicular to any root. </p><p>The <b>E<sub>8</sub> root system</b> is a rank 8 root system containing 240 root vectors spanning <b>R</b><sup>8</sup>. It is <a href="/wiki/Irreducibility_(mathematics)" title="Irreducibility (mathematics)">irreducible</a> in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E<sub>8</sub> have the same length. It is convenient for a number of purposes to normalize them to have length <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">2</span></span>. These 240 vectors are the vertices of a <a href="/wiki/Semi-regular_polytope" class="mw-redirect" title="Semi-regular polytope">semi-regular polytope</a> discovered by <a href="/wiki/Thorold_Gosset" title="Thorold Gosset">Thorold Gosset</a> in 1900, sometimes known as the <a href="/wiki/4_21_polytope" title="4 21 polytope">4<sub>21</sub> polytope</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Construction">Construction</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=7" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the so-called <i>even coordinate system</i>, E<sub>8</sub> is given as the set of all vectors in <b>R</b><sup>8</sup> with length squared equal to 2 such that coordinates are either all <a href="/wiki/Integer" title="Integer">integers</a> or all <a href="/wiki/Half-integer" title="Half-integer">half-integers</a> and the sum of the coordinates is even. </p><p>Explicitly, there are 112 roots with integer entries obtained from </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 \left(\pm 1,\pm 1,0,0,0,0,0,0\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\pm 1,\pm 1,0,0,0,0,0,0\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee898bb9ac7c01c57db613cbeb2579a9208166f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.35ex; height:2.843ex;" alt="{\displaystyle \left(\pm 1,\pm 1,0,0,0,0,0,0\right)\,}"></span></dd></dl> <p>by taking an arbitrary combination of signs and an arbitrary <a href="/wiki/Permutation" title="Permutation">permutation</a> of coordinates, and 128 roots with half-integer entries obtained from </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 \left(\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8c17bae1db82304956f44f9dd196eb796c6f91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.484ex; height:3.509ex;" alt="{\displaystyle \left(\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\right)\,}"></span></dd></dl> <p>by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:E8-with-thread.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/E8-with-thread.jpg/220px-E8-with-thread.jpg" decoding="async" width="220" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/E8-with-thread.jpg/330px-E8-with-thread.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/37/E8-with-thread.jpg/440px-E8-with-thread.jpg 2x" data-file-width="1700" data-file-height="1499" /></a><figcaption>E<sub>8</sub> 2d projection with thread made by hand</figcaption></figure> <p>The 112 roots with integer entries form a D<sub>8</sub> root system. The E<sub>8</sub> root system also contains a copy of A<sub>8</sub> (which has 72 roots) as well as <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a> and <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a> (in fact, the latter two are usually <i>defined</i> as subsets of E<sub>8</sub>). </p><p>In the <i>odd coordinate system</i>, E<sub>8</sub> is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number. </p> <div class="mw-heading mw-heading3"><h3 id="Dynkin_diagram">Dynkin diagram</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=8" title="Edit section: Dynkin diagram"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagram</a> for E<sub>8</sub> is given by <span typeof="mw:File"><a href="/wiki/File:Dynkin_diagram_E8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Dynkin_diagram_E8.svg/120px-Dynkin_diagram_E8.svg.png" decoding="async" width="120" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Dynkin_diagram_E8.svg/180px-Dynkin_diagram_E8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e8/Dynkin_diagram_E8.svg/240px-Dynkin_diagram_E8.svg.png 2x" data-file-width="389" data-file-height="91" /></a></span>. </p><p>This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots that are not joined by a line are <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Cartan_matrix">Cartan matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=9" title="Edit section: Cartan matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan matrix</a> of a rank <span class="texhtml"><i>r</i></span> root system is an <span class="texhtml"><i>r × r</i></span> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by </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_{ij}=2{\frac {\left(\alpha _{i},\alpha _{j}\right)}{\left(\alpha _{i},\alpha _{i}\right)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}=2{\frac {\left(\alpha _{i},\alpha _{j}\right)}{\left(\alpha _{i},\alpha _{i}\right)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/109debe85772bb5097ef1a6776556312e76b8320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:15.845ex; height:6.509ex;" alt="{\displaystyle A_{ij}=2{\frac {\left(\alpha _{i},\alpha _{j}\right)}{\left(\alpha _{i},\alpha _{i}\right)}}}"></span></dd></dl> <p>where <span class="texhtml">( , )</span> is the Euclidean <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> and <span class="texhtml"><i>α<sub>i</sub></i></span> are the simple roots. The entries are independent of the choice of simple roots (up to ordering). </p><p>The Cartan matrix for E<sub>8</sub> is given by </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 \left[{\begin{array}{rr}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&-1\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&0&0&-1&0&0&2\end{array}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{rr}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&-1\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&0&0&-1&0&0&2\end{array}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/744a9f7c0ba8232d9458e553177042d90ff6ebd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:44.909ex; height:25.509ex;" alt="{\displaystyle \left[{\begin{array}{rr}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&-1\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&0&0&-1&0&0&2\end{array}}\right].}"></span></dd></dl> <p>The <a href="/wiki/Determinant" title="Determinant">determinant</a> of this matrix is equal to 1. </p> <div class="mw-heading mw-heading3"><h3 id="Simple_roots">Simple roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=10" title="Edit section: Simple roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:E8HassePoset.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/E8HassePoset.svg/170px-E8HassePoset.svg.png" decoding="async" width="170" height="481" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/E8HassePoset.svg/255px-E8HassePoset.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b2/E8HassePoset.svg/340px-E8HassePoset.svg.png 2x" data-file-width="1208" data-file-height="3416" /></a><figcaption><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of E<sub>8</sub> <a href="/wiki/Root_system#The_root_poset" title="Root system">root poset</a> with edge labels identifying added simple root position</figcaption></figure> <p>A set of <a href="/wiki/Simple_root_(root_system)" class="mw-redirect" title="Simple root (root system)">simple roots</a> for a root system Φ is a set of roots that form a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive. </p><p>Given the E<sub>8</sub> <a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan matrix</a> (above) and a <a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagram</a> node ordering of: <span typeof="mw:File"><a href="/wiki/File:DynkinE8.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/DynkinE8.svg/150px-DynkinE8.svg.png" decoding="async" width="150" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/DynkinE8.svg/225px-DynkinE8.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/DynkinE8.svg/300px-DynkinE8.svg.png 2x" data-file-width="425" data-file-height="105" /></a></span> </p><p>One choice of <a href="/wiki/Simple_root_(root_system)" class="mw-redirect" title="Simple root (root system)">simple roots</a> is given by the rows of the following matrix: </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 \left[{\begin{array}{rr}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&1&0\\-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&0&0&0&0&1&-1&0\\\end{array}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{rr}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&1&0\\-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&0&0&0&0&1&-1&0\\\end{array}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ced7ddba33a94a3b3e33d95c8b34d1dd94200e01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.671ex; width:48.874ex; height:26.509ex;" alt="{\displaystyle \left[{\begin{array}{rr}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&1&0\\-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&0&0&0&0&1&-1&0\\\end{array}}\right].}"></span></dd></dl> <p>With this numbering of nodes in the Dynkin diagram, the highest root in the root system has <a href="/w/index.php?title=Coxeter_label&action=edit&redlink=1" class="new" title="Coxeter label (page does not exist)">Coxeter labels</a> (2, 3, 4, 5, 6, 4, 2, 3). Using this representation of the simple roots, the lowest root is given by </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 \left[{\begin{array}{rr}-1&0&0&0&0&0&0&1\\\end{array}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{rr}-1&0&0&0&0&0&0&1\\\end{array}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe60dcef180bbd3b7a5ae0d5cfa9c94a7fc3a3f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.445ex; height:2.843ex;" alt="{\displaystyle \left[{\begin{array}{rr}-1&0&0&0&0&0&0&1\\\end{array}}\right].}"></span></dd></dl> <p>The only simple root that can be added to the lowest root to obtain another root is the one corresponding to node 1 in this labeling of the Dynkin diagram — as is to be expected from the <a href="/wiki/Dynkin_diagram#Affine_Dynkin_diagrams" title="Dynkin diagram">affine Dynkin diagram</a> 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 {\tilde {\mathrm {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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> <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 {\mathrm {E} }}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e22bd6e109d73905e33bdefe5d32b6aa81ed50d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.637ex; height:3.009ex;" alt="{\displaystyle {\tilde {\mathrm {E} }}_{8}}"></span>. The <a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> to the right enumerates the 120 roots of positive height relative to any particular choice of simple roots consistent with this node numbering. </p><p>Note that the Hasse diagram does not represent the full Lie algebra, or even the full root system. The 120 roots of negative height relative to the same set of simple roots can be adequately represented by a second copy of the Hasse diagram with the arrows reversed; but it is less straightforward to connect these two diagrams via a basis for the eight-dimensional Cartan subalgebra. In the notation of <a href="/wiki/Semisimple_Lie_algebra#Structure" title="Semisimple Lie algebra">the exposition of Chevalley generators and Serre relations</a>: Insofar as an arrow represents the Lie bracket by the generator <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}"></span> associated with a simple root, each root in the height -1 layer of the reversed Hasse diagram must correspond to some <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span> and can have only one upward arrow, connected to a node in the height 0 layer representing the element of the Cartan subalgebra given by <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_{i}=[e_{i},f_{i}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{i}=[e_{i},f_{i}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dab22318667163fed6ffa024e12c7511763d312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.387ex; height:2.843ex;" alt="{\displaystyle h_{i}=[e_{i},f_{i}]}"></span>. But the upward arrows from the height 0 layer must then represent <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_{i},h_{j}]=-a_{ji}e_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [e_{i},h_{j}]=-a_{ji}e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70c091066e333cb11e80abc1881690737b2179b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.956ex; height:3.009ex;" alt="{\displaystyle [e_{i},h_{j}]=-a_{ji}e_{i}}"></span>, where <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_{ji}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3630693f8dd03d56a80ad9b4fde40e3e14020962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ji}}"></span> is (the transpose of) the Cartan matrix. One could draw multiple upward arrows from each <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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/614689f15f73ad5b4a5d7fa837a72614202b0d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.249ex; height:2.843ex;" alt="{\displaystyle h_{j}}"></span> associated with all <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}"></span> for which <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_{i},h_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [e_{i},h_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/418b426256d20d8a6ea4c47bd1623b53cb7c9845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.459ex; height:3.009ex;" alt="{\displaystyle [e_{i},h_{j}]}"></span> is nonzero; but this neither captures the numerical entries in the Cartan matrix nor reflects the fact that each <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}"></span> only has nonzero Lie bracket with one degree of freedom in the Cartan subalgebra (just not the same degree of freedom 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 h_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d535f210cbd9b9fe6689e61427b3e213e5b2d547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.139ex; height:2.509ex;" alt="{\displaystyle h_{i}}"></span>). </p><p>More fundamentally, this organization implies that the <i>span</i> of the generators designated as "the" Cartan subalgebra is somehow inherently special, when in most applications, any mutually commuting set of eight of the 248 Lie algebra generators (of which there are many!) — or any eight linearly independent, mutually commuting Lie derivations on any manifold with E<sub>8</sub> structure — would have served just as well. Once a Cartan subalgebra has been selected (or defined <i>a priori</i>, as in the case of a lattice), a basis of "Cartan generators" (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 h_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d535f210cbd9b9fe6689e61427b3e213e5b2d547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.139ex; height:2.509ex;" alt="{\displaystyle h_{i}}"></span> among the Chevalley generators) and a root system are a useful way to describe structure <i>relative to this subalgebra</i>. But the root system map is not the Lie algebra (let alone group!) territory. Given a set of Chevalley generators, most degrees of freedom in a Lie algebra and their sparse Lie brackets with <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_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {e_{i}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f39da2d23ba0eb794199fff12f4962555a2275c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle {e_{i}}}"></span> can be represented schematically as circles and arrows, but this simply breaks down on the chosen Cartan subalgebra. Such are the hazards of schematic visual representations of mathematical structures. </p> <div class="mw-heading mw-heading3"><h3 id="Weyl_group">Weyl group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=11" title="Edit section: Weyl group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Weyl_group" title="Weyl group">Weyl group</a> of E<sub>8</sub> is of order 696729600, and can be described as O<span class="nowrap"><span style="display:inline-block;margin-bottom:-0.3em;vertical-align:-0.4em;line-height:1.2em;font-size:80%;text-align:left"><sup style="font-size:inherit;line-height:inherit;vertical-align:baseline">+</sup><br /><sub style="font-size:inherit;line-height:inherit;vertical-align:baseline">8</sub></span></span>(2): it is of the form 2.<i>G</i>.2 (that is, a <a href="/wiki/Schur_multiplier" title="Schur multiplier">stem extension</a> by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group <i>G</i>) where <i>G</i> is the unique <a href="/wiki/Simple_group" title="Simple group">simple group</a> of order 174182400 (which can be described as PSΩ<sub>8</sub><sup>+</sup>(2)).<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> </p> <div class="mw-heading mw-heading3"><h3 id="E8_root_lattice">E<sub>8</sub> root lattice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=12" title="Edit section: E8 root lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/E8_lattice" title="E8 lattice">E<sub>8</sub> lattice</a></div> <p>The integral span of the E<sub>8</sub> root system forms a <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> in <b>R</b><sup>8</sup> naturally called the <b><a href="/wiki/E8_lattice" title="E8 lattice">E<sub>8</sub> root lattice</a></b>. This lattice is rather remarkable in that it is the only (nontrivial) even, <a href="/wiki/Unimodular_lattice" title="Unimodular lattice">unimodular lattice</a> with rank less than 16. </p> <div class="mw-heading mw-heading3"><h3 id="Simple_subalgebras_of_E8">Simple subalgebras of E<sub>8</sub></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=13" title="Edit section: Simple subalgebras of E8"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:E8SubgroupTree.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/E8SubgroupTree.svg/220px-E8SubgroupTree.svg.png" decoding="async" width="220" height="607" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/E8SubgroupTree.svg/330px-E8SubgroupTree.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/E8SubgroupTree.svg/440px-E8SubgroupTree.svg.png 2x" data-file-width="967" data-file-height="2667" /></a><figcaption>An incomplete simple subgroup tree of E<sub>8</sub></figcaption></figure> <p>The Lie algebra E<sub>8</sub> contains as subalgebras all the <a href="/wiki/Exceptional_Lie_algebra" title="Exceptional Lie algebra">exceptional Lie algebras</a> as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra. </p> <div class="mw-heading mw-heading2"><h2 id="Chevalley_groups_of_type_E8">Chevalley groups of type E<sub>8</sub></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=14" title="Edit section: Chevalley groups of type E8"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="#CITEREFChevalley1955">Chevalley (1955)</a> showed that the points of the (split) algebraic group E<sub>8</sub> (see <a href="#E8_as_an_algebraic_group">above</a>) over a <a href="/wiki/Finite_field" title="Finite field">finite field</a> with <i>q</i> elements form a finite <a href="/wiki/Group_of_Lie_type" title="Group of Lie type">Chevalley group</a>, generally written E<sub>8</sub>(<i>q</i>), which is simple for any <i>q</i>,<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><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> and constitutes one of the infinite families addressed by the <a href="/wiki/Classification_of_finite_simple_groups" title="Classification of finite simple groups">classification of finite simple groups</a>. Its number of elements is given by the formula (sequence <span class="nowrap external"><a href="//oeis.org/A008868" class="extiw" title="oeis:A008868">A008868</a></span> in the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</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 q^{120}\left(q^{30}-1\right)\left(q^{24}-1\right)\left(q^{20}-1\right)\left(q^{18}-1\right)\left(q^{14}-1\right)\left(q^{12}-1\right)\left(q^{8}-1\right)\left(q^{2}-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>120</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>30</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q^{120}\left(q^{30}-1\right)\left(q^{24}-1\right)\left(q^{20}-1\right)\left(q^{18}-1\right)\left(q^{14}-1\right)\left(q^{12}-1\right)\left(q^{8}-1\right)\left(q^{2}-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6dec5674c68c66f1f71c86995b471da9f5796c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:77.937ex; height:3.343ex;" alt="{\displaystyle q^{120}\left(q^{30}-1\right)\left(q^{24}-1\right)\left(q^{20}-1\right)\left(q^{18}-1\right)\left(q^{14}-1\right)\left(q^{12}-1\right)\left(q^{8}-1\right)\left(q^{2}-1\right)}"></span></dd></dl> <p>The first term in this sequence, the order of E<sub>8</sub>(2), namely <span style="white-space:nowrap">337<span style="margin-left:0.25em">804</span><span style="margin-left:0.25em">753</span><span style="margin-left:0.25em">143</span><span style="margin-left:0.25em">634</span><span style="margin-left:0.25em">806</span><span style="margin-left:0.25em">261</span><span style="margin-left:0.25em">388</span><span style="margin-left:0.25em">190</span><span style="margin-left:0.25em">614</span><span style="margin-left:0.25em">085</span><span style="margin-left:0.25em">595</span><span style="margin-left:0.25em">079</span><span style="margin-left:0.25em">991</span><span style="margin-left:0.25em">692</span><span style="margin-left:0.25em">242</span><span style="margin-left:0.25em">467</span><span style="margin-left:0.25em">651</span><span style="margin-left:0.25em">576</span><span style="margin-left:0.25em">160</span><span style="margin-left:0.25em">959</span><span style="margin-left:0.25em">909</span><span style="margin-left:0.25em">068</span><span style="margin-left:0.25em">800</span><span style="margin-left:0.25em">000</span></span> ≈ <span class="nowrap"><span data-sort-value="7074337999999999999♠"></span>3.38<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>74</sup></span>, is already larger than the size of the <a href="/wiki/Monster_group" title="Monster group">Monster group</a>. This group E<sub>8</sub>(2) is the last one described (but without its character table) in the <a href="/wiki/ATLAS_of_Finite_Groups" title="ATLAS of Finite Groups">ATLAS of Finite Groups</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> </p><p>The <a href="/wiki/Schur_multiplier" title="Schur multiplier">Schur multiplier</a> of E<sub>8</sub>(<i>q</i>) is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order <i>f</i> if <span class="nowrap"><i>q</i> = <i>p<sup>f</sup></i></span>, where <i>p</i> is prime). </p><p><a href="#CITEREFLusztig1979">Lusztig (1979)</a> described the unipotent representations of finite groups of type <i>E</i><sub>8</sub>. </p> <div class="mw-heading mw-heading2"><h2 id="Subgroups,_subalgebras,_and_extensions"><span id="Subgroups.2C_subalgebras.2C_and_extensions"></span>Subgroups, subalgebras, and extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=15" title="Edit section: Subgroups, subalgebras, and extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The smaller exceptional groups <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a> and <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a> sit inside E<sub>8</sub>. In the compact group, both E<sub>6</sub>×SU(3)/(<b>Z</b> / 3<b>Z</b>) and E<sub>7</sub>×SU(2)/(+1,−1) are <a href="/wiki/Maximal_subgroup" title="Maximal subgroup">maximal subgroups</a> of E<sub>8</sub>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:E8_Maximal_Embeddings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/E8_Maximal_Embeddings.svg/300px-E8_Maximal_Embeddings.svg.png" decoding="async" width="300" height="684" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/E8_Maximal_Embeddings.svg/450px-E8_Maximal_Embeddings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d6/E8_Maximal_Embeddings.svg/600px-E8_Maximal_Embeddings.svg.png 2x" data-file-width="1320" data-file-height="3009" /></a><figcaption>Embeddings of the maximal subgroups of E<sub>8</sub> up to dimension 248 with associated projection matrix.</figcaption></figure> <p>The 248-dimensional adjoint representation of E<sub>8</sub> may be considered in terms of its <a href="/wiki/Restricted_representation" title="Restricted representation">restricted representation</a> to the first of these subgroups. It transforms under E<sub>6</sub>×SU(3) as a sum of <a href="/wiki/Tensor_product_representation" class="mw-redirect" title="Tensor product representation">tensor product representations</a>, which may be labelled as a pair of dimensions as (78,1) + (1,8) + (<span style="text-decoration:overline;">27</span>,3) + (27,<span style="text-decoration:overline;">3</span>. (Since the maximal subgroup is actually the quotient of this group product by a finite group, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations.) Since the adjoint representation can be described by the roots together with the generators in the <a href="/wiki/Cartan_subalgebra" title="Cartan subalgebra">Cartan subalgebra</a>, we may choose a particular E<sub>6</sub> root system within E<sub>8</sub> and decompose the sum representation relative to this E<sub>6</sub>. In this description, </p> <ul><li>(78,1), a copy 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 {\mathfrak {e}}_{6}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f478652f321cdcdc6a3a0185a7beac54022adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{6}}"></span>, consists of the 72 roots with (−<style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), (0,0,0), or (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>) in the last three dimensions, together with five Cartan generators corresponding to the first five dimensions and one Cartan generator corresponding to an equally weighted sum of the last three Cartan generators of the E<sub>8</sub> system;</li> <li>(1,8), a copy 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 {\mathfrak {su}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e67744b88c1161065934d1d841d185d6a7bb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(3)}"></span>, consists of the six roots with permutations of (0,1,−1) in the last three dimensions, together with two Cartan generators corresponding to the two trace-free combinations of the last three Cartan generators of the E<sub>8</sub> system;</li> <li>(<u style="text-decoration:overline">27</u>,3) consists of all roots with permutations of (-1,0,0), (−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), or (0,1,1) in the last three dimensions; and</li> <li>(27,<u style="text-decoration:overline">3</u>) consists of all roots with permutations of (1,0,0), (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), or (0,-1,-1) in the last three dimensions.</li></ul> <p>The 248-dimensional adjoint representation of E<sub>8</sub>, when similarly restricted to the second maximal subgroup, transforms under E<sub>7</sub>×SU(2) as: (133,1) + (1,3) + (56,2). We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description, </p> <ul><li>(133,1), a copy 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 {\mathfrak {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span>, consists of the 126 roots with (−1,−1), (−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), (0,0), (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), or (1,1) in the last two dimensions, together with six Cartan generators corresponding to the first six dimensions and one Cartan generator corresponding to an equally weighted sum of the last two Cartan generators of the E<sub>8</sub> system;</li> <li>(1,3), a copy 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 {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"></span>, consists of the two roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1), together with one Cartan generator corresponding to the trace-free combination of the last two Cartan generators of the E<sub>8</sub> system; and</li> <li>(56,2) consists of all roots with permutations of (0,-1), (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), or (1,0) in the last two dimensions.</li></ul> <p>The connection between these two descriptions is given by the graded exceptional Lie algebra constructions of <a href="/wiki/Kantor%E2%80%93Koecher%E2%80%93Tits_construction" title="Kantor–Koecher–Tits construction">J. Tits</a> and <a href="/wiki/Structurable_algebra" title="Structurable algebra">B. N. Allison</a>. Any 27-dimensional representation of E<sub>6</sub> can be equipped with a non-associative (but strictly <a href="/wiki/Power-associative" class="mw-redirect" title="Power-associative">power-associative</a>) Jordan product operation to form an <a href="/wiki/Albert_algebra" title="Albert algebra">Albert algebra</a> (an important exceptional case in <a rel="nofollow" class="external text" href="https://www.math.uni-bielefeld.de/LAG/man/089.pdf">algebraic constructions</a>). The <a href="/wiki/Kantor%E2%80%93Koecher%E2%80%93Tits_construction" title="Kantor–Koecher–Tits construction">Kantor–Koecher–Tits construction</a> applied to this Albert algebra recovers the 78-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 {\mathfrak {e}}_{6}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f478652f321cdcdc6a3a0185a7beac54022adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{6}}"></span> as the reduced structure algebra of the Albert algebra. This <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 {e}}_{6}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f478652f321cdcdc6a3a0185a7beac54022adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{6}}"></span>, together with the 27 and <u style="text-decoration:overline">27</u> representations and the grade operator (the element of the Cartan subalgebra with weight -1 on the 27, +1 on the <u style="text-decoration:overline">27</u>, and 0 on the 78), forms an <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 {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span> 3-graded Lie algebra. A complete exposition of this construction may be found in standard texts on Jordan algebras such as <a rel="nofollow" class="external text" href="https://www.worldcat.org/title/structure-and-representations-of-jordan-algebras/oclc/1077">Jacobson 1968</a> or <a rel="nofollow" class="external text" href="http://old.math.nsc.ru/LBRT/a1/files/mccrimmon.pdf">McCrimmon 2004</a>. </p><p>Starting this 3-graded Lie algebra construction with any particular 27-dimensional representation, embedded within <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 {e}}_{8}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ae581bd92e06f5a31fb26d91f745d71941fc91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{8}}"></span>, of any particular E<sub>6</sub> subgroup of E<sub>8</sub> produces the corresponding <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 {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span> subalgebra. The particular <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 {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span> in the E<sub>7</sub>×SU(2) decomposition given above corresponds to choosing the 27 consisting of all roots with (1,0,0), (<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), or (0,-1,-1) in the last three dimensions (in order), with the grade operator having weight (-1,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>) in these dimensions; or equivalently to choosing the "<u style="text-decoration:overline">27</u>" consisting of all roots with (-1,0,0), (−<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>), or (0,1,1) in the last three dimensions (in order), with the grade operator having weight (1,-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>,-<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>) in these dimensions. Note that there is nothing special about this choice of dimensions — 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 \mathbb {R} ^{8}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ccc2412bb86d0c9bf29d71756465f9353bec870" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{8}}"></span> within which the root system is embedded is not the set of eight independent but non-orthogonal axes corresponding to the simple roots, and any three dimensions will do — and there are also constructions using other equivalent groupings of roots. What matters is that the kernel of the Lie bracket with the generator chosen as the "grade operator" be an <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 {e}}_{6}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f478652f321cdcdc6a3a0185a7beac54022adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{6}}"></span> subalgebra (plus a central <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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> associated with the grade operator itself and the remaining generator of the Cartan subalgebra), not <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 {a}}_{6}\oplus \mathbb {R} ^{2}}"> <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">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo>⊕</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{6}\oplus \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45240d75e8877a7b6fcbb35b8dc6a1c524c2b785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.789ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {a}}_{6}\oplus \mathbb {R} ^{2}}"></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 {\mathfrak {a}}_{7}\oplus \mathbb {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">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{7}\oplus \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac2ecf0b2a9b47649a8f552b4297a4794f97ace4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.735ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}_{7}\oplus \mathbb {R} }"></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 {\mathfrak {e}}_{7}\oplus \mathbb {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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}\oplus \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd31387f1a7002fd65007517c803c14d711c230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.505ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {e}}_{7}\oplus \mathbb {R} }"></span>, <i>etc.</i> </p><p>(In the simple Lie algebra case, the sign of the grade of the 27 versus the <u style="text-decoration:overline">27</u> representation is a matter of convention, as is the scale of the grading. However, choosing -1 as the grade of the 27-dimensional "vector" representation is consistent with an extension of the 3-graded algebra to higher positive grades via the exterior algebra over the <u style="text-decoration:overline">27</u> "covector" representation. The "vector" representation then lies, not in this nonnegative-graded exterior algebra, but in the graded algebra of derivations over the exterior algebra; the 78-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 {\mathfrak {e}}_{6}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f478652f321cdcdc6a3a0185a7beac54022adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{6}}"></span> is the grade 0 subalgebra (a subalgebra of the inner derivations by "vector-valued 1-forms") of this graded algebra of derivations. For details on how this asymmetric structure works starting from a general 3-graded algebra, see references at <a href="/wiki/Fr%C3%B6licher%E2%80%93Nijenhuis_bracket" title="Frölicher–Nijenhuis bracket">Frölicher–Nijenhuis bracket</a>. The relevance of this observation to E<sub>8</sub> is simply that E<sub>7</sub> and E<sub>8</sub> are their own clusters of structures, distinguished as exceptional simple Lie groups/algebras, and that any particular reconstruction of them using representations of their subgroups/subalgebras will have extensions beyond the motivating case. Varying conventions of sign, scale, and conjugate relationship in the literature are due not just to inaccuracies but also to the directions in which the authors seek to extend their constructions.) </p><p>The distinguished <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}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"></span> in the E<sub>7</sub>×SU(2) decomposition above is then given by the subalgebra 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 {\mathfrak {su}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e67744b88c1161065934d1d841d185d6a7bb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(3)}"></span> that commutes with the grade operator (which lies in the Cartan subalgebra of this <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}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e67744b88c1161065934d1d841d185d6a7bb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(3)}"></span>). Of the four remaining roots in 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 {\mathfrak {su}}(3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62e67744b88c1161065934d1d841d185d6a7bb7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(3)}"></span>, two are of grade <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> and two are of grade -<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>. In the convention where the 27 of E<sub>6</sub> used to construct 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 {\mathfrak {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span> has grade -1 and the <u style="text-decoration:overline">27</u> has grade +1, the other two 27's have grade +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> and the other two <u style="text-decoration:overline">27</u>'s have grade -<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span>, as is apparent from permuting the values of the last three roots in the description above. Grouping these 4×(1+27)=112 generators to form the grade +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> and -<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> subspaces 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 {\mathfrak {e}}_{8}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ae581bd92e06f5a31fb26d91f745d71941fc91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{8}}"></span> (relative to the original choice of grade operator within <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 {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span>), each subspace may be given a quite particular non-associative (nor even power-associative) product operation, resulting in two copies of <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/71948">Brown's 56-dimensional structurable algebra</a>. Allison's 5-graded Lie algebra construction based on this <a href="/wiki/Structurable_algebra" title="Structurable algebra">structurable algebra</a> recovers the original <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 {e}}_{8}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ae581bd92e06f5a31fb26d91f745d71941fc91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{8}}"></span>. (Allison's 5-grading differs from the above by a factor -2.) Grouping these generators differently, based on their weights relative to the Cartan generator 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 {\mathfrak {su}}(2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">s</mi> <mi mathvariant="fraktur">u</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {su}}(2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369267d5eae15a7478ebb52c71fcec1449a10dc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.041ex; width:5.244ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {su}}(2)}"></span> <i>orthogonal</i> 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 {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span>, gives two 56-dimensional subspaces that each carry the lowest-dimensional non-trivial irreducible representation of E<sub>7</sub>. Either of these may be combined with the Cartan generator to form a 57-dimensional <a href="/wiki/Heisenberg_algebra" class="mw-redirect" title="Heisenberg algebra">Heisenberg algebra</a>, and adjoining this 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 {e}}_{7}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{7}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f344b465718da83d20783caafaa659afc61ad644" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{7}}"></span> produces the (non-simple) Lie algebra <a href="/wiki/E7_1/2" class="mw-redirect" title="E7 1/2">E<sub>7 1/2</sub></a> described by Landsberg and Manivel. </p><p>From the perspective in which the 27-dimensional grade -1 subspace 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 {\mathfrak {e}}_{8}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ae581bd92e06f5a31fb26d91f745d71941fc91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{8}}"></span> (relative to a choice of grade operator) plays the role of "vector" representation of E<sub>6</sub> and the <u style="text-decoration:overline">27</u> with roots opposite it plays the role of "covector" representation, it is natural to look for "spinor" representations in the grade +<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> and -<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027"><span class="frac"><span class="num">1</span>⁄<span class="den">2</span></span> subspaces, or in some other combination of the (<u style="text-decoration:overline">27</u>,3) and (27,<u style="text-decoration:overline">3</u>) representations of E<sub>6</sub>×SU(3), and to attempt to relate these to geometrical spinors in the <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> sense as employed in <a href="/wiki/Quantum_field_theory" title="Quantum field theory">quantum field theory</a>. Variations on this idea are <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/2005.03048.pdf">common in the physics literature</a>. See <a rel="nofollow" class="external text" href="https://arxiv.org/pdf/0905.2658.pdf">Distler and Garibaldi 2009</a> for discussion of the mathematical obstacles to constructing a <i>chiral</i> gauge theory based on E<sub>8</sub>. The structure 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 {\mathfrak {e}}_{8}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04ae581bd92e06f5a31fb26d91f745d71941fc91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{8}}"></span> relative to its <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 {e}}_{6}}"> <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">e</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {e}}_{6}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f478652f321cdcdc6a3a0185a7beac54022adb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.987ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {e}}_{6}}"></span> subalgebra, together with the conventional scaling of elements of the Cartan subalgebra, invites extensions by geometric analogy but does not necessarily imply a relationship to low-dimensional geometry or low-energy physics. The same may be said of connections to Jordan and Heisenberg algebras, whose historical origins are intertwined with the development of quantum mechanics. Not every <a href="/wiki/The_Treachery_of_Images" title="The Treachery of Images">visual representation evocative of a tobacco pipe</a> will hold tobacco. </p><p>The finite quasisimple groups that can embed in (the compact form of) E<sub>8</sub> were found by <a href="#CITEREFGriessRyba1999">Griess & Ryba (1999)</a>. </p><p>The <a href="/wiki/Dempwolff_group" title="Dempwolff group">Dempwolff group</a> is a subgroup of (the compact form of) E<sub>8</sub>. It is contained in the <a href="/wiki/Thompson_sporadic_group" title="Thompson sporadic group">Thompson sporadic group</a>, which acts on the underlying vector space of the Lie group E<sub>8</sub> but does not preserve the Lie bracket. The Thompson group fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompson group into E<sub>8</sub>(<b>F</b><sub>3</sub>). </p><p>The embeddings of the maximal subgroups of E<sub>8</sub> up to dimension 248 are shown to the right. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=16" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The E<sub>8</sub> Lie group has applications in <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> and especially in <a href="/wiki/String_theory" title="String theory">string theory</a> and <a href="/wiki/Supergravity" title="Supergravity">supergravity</a>. E<sub>8</sub>×E<sub>8</sub> is the <a href="/wiki/Gauge_group" class="mw-redirect" title="Gauge group">gauge group</a> of one of the two types of <a href="/wiki/Heterotic_string" class="mw-redirect" title="Heterotic string">heterotic string</a> and is one of two <a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">anomaly-free</a> gauge groups that can be coupled to the <i>N</i> = 1 supergravity in ten dimensions. E<sub>8</sub> is the <a href="/wiki/U-duality" title="U-duality">U-duality</a> group of supergravity on an eight-torus (in its split form). </p><p>One way to incorporate the <a href="/wiki/Standard_model" class="mw-redirect" title="Standard model">standard model</a> of particle physics into heterotic string theory is the <a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">symmetry breaking</a> of E<sub>8</sub> to its maximal subalgebra SU(3)×E<sub>6</sub>. </p><p>In 1982, <a href="/wiki/Michael_Freedman" title="Michael Freedman">Michael Freedman</a> used the <a href="/wiki/E8_lattice" title="E8 lattice">E<sub>8</sub> lattice</a> to construct an example of a <a href="/wiki/Topological_manifold" title="Topological manifold">topological</a> <a href="/wiki/4-manifold" title="4-manifold">4-manifold</a>, the <a href="/wiki/E8_manifold" title="E8 manifold">E<sub>8</sub> manifold</a>, which has no <a href="/wiki/Differential_structure" title="Differential structure">smooth structure</a>. </p><p><a href="/wiki/Antony_Garrett_Lisi" title="Antony Garrett Lisi">Antony Garrett Lisi</a>'s incomplete "<a href="/wiki/An_Exceptionally_Simple_Theory_of_Everything" title="An Exceptionally Simple Theory of Everything">An Exceptionally Simple Theory of Everything</a>" attempts to describe all known <a href="/wiki/Fundamental_interaction" title="Fundamental interaction">fundamental interactions</a> in physics as part of the E<sub>8</sub> Lie algebra.<sup id="cite_ref-SciAm_7-0" class="reference"><a href="#cite_note-SciAm-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-seed_8-0" class="reference"><a href="#cite_note-seed-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (<a href="#CITEREFColdeaTennantWheelerWawrzynska2010">2010</a>) reported an experiment where the <a href="/wiki/Electron_spin" class="mw-redirect" title="Electron spin">electron spins</a> of a <a href="/wiki/Cobalt" title="Cobalt">cobalt</a>-<a href="/wiki/Niobium" title="Niobium">niobium</a> crystal exhibited, under certain conditions, two of the eight peaks related to E<sub>8</sub> that were predicted by <a href="#CITEREFZamolodchikov1989">Zamolodchikov (1989)</a>.<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-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=17" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Wilhelm_Killing" title="Wilhelm Killing">Wilhelm Killing</a> (<a href="#CITEREFKilling1888a">1888a</a>, <a href="#CITEREFKilling1888b">1888b</a>, <a href="#CITEREFKilling1889">1889</a>, <a href="#CITEREFKilling1890">1890</a>) discovered the complex Lie algebra E<sub>8</sub> during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by <a href="/wiki/%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a>. Cartan determined that a complex simple Lie algebra of type E<sub>8</sub> admits three real forms. Each of them gives rise to a simple <a href="/wiki/Lie_group" title="Lie group">Lie group</a> of dimension 248, exactly one of which (as for any complex simple Lie algebra) is <a href="/wiki/Compact_Lie_group" class="mw-redirect" title="Compact Lie group">compact</a>. <a href="#CITEREFChevalley1955">Chevalley (1955)</a> introduced <a href="/wiki/Algebraic_group" title="Algebraic group">algebraic groups</a> and Lie algebras of type E<sub>8</sub> over other <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>: for example, in the case of <a href="/wiki/Finite_field" title="Finite field">finite fields</a> they lead to an infinite family of <a href="/wiki/Finite_simple_group" class="mw-redirect" title="Finite simple group">finite simple groups</a> of Lie type. E<sub>8</sub> continues to be an area of active basic research by <a href="/wiki/Atlas_of_Lie_Groups_and_Representations" class="mw-redirect" title="Atlas of Lie Groups and Representations">Atlas of Lie Groups and Representations</a>, which aims to determine the unitary representations of all the Lie groups.<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> </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=E8_(mathematics)&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/En_(Lie_algebra)" title="En (Lie algebra)">E<sub><i>n</i></sub></a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=19" title="Edit section: Footnotes"><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"><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="CITEREFПлатоновРапинчук1991" class="citation cs2">Платонов, Владимир П.; Рапинчук, Андрей С. (1991), <i>Алгебраические группы и теория чисел</i>, Наука, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/5-02-014191-7" title="Special:BookSources/5-02-014191-7"><bdi>5-02-014191-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B5+%D0%B3%D1%80%D1%83%D0%BF%D0%BF%D1%8B+%D0%B8+%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F+%D1%87%D0%B8%D1%81%D0%B5%D0%BB&rft.pub=%D0%9D%D0%B0%D1%83%D0%BA%D0%B0&rft.date=1991&rft.isbn=5-02-014191-7&rft.aulast=%D0%9F%D0%BB%D0%B0%D1%82%D0%BE%D0%BD%D0%BE%D0%B2&rft.aufirst=%D0%92%D0%BB%D0%B0%D0%B4%D0%B8%D0%BC%D0%B8%D1%80+%D0%9F.&rft.au=%D0%A0%D0%B0%D0%BF%D0%B8%D0%BD%D1%87%D1%83%D0%BA%2C+%D0%90%D0%BD%D0%B4%D1%80%D0%B5%D0%B9+%D0%A1.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span> (English translation: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlatonovRapinchuk1994" class="citation cs2">Platonov, Vladimir P.; Rapinchuk, Andrei S. (1994), <i>Algebraic groups and number theory</i>, Academic Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-12-558180-7" title="Special:BookSources/0-12-558180-7"><bdi>0-12-558180-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+groups+and+number+theory&rft.pub=Academic+Press&rft.date=1994&rft.isbn=0-12-558180-7&rft.aulast=Platonov&rft.aufirst=Vladimir+P.&rft.au=Rapinchuk%2C+Andrei+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span>), §2.2.4</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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://johncarlosbaez.wordpress.com/2017/12/16/the-600-cell/">"The 600-Cell (Part 1)"</a>. December 16, 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+600-Cell+%28Part+1%29&rft.date=2017-12-16&rft_id=https%3A%2F%2Fjohncarlosbaez.wordpress.com%2F2017%2F12%2F16%2Fthe-600-cell%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFConwayCurtisNortonParker1985" class="citation cs2"><a href="/wiki/John_Horton_Conway" title="John Horton Conway">Conway, John Horton</a>; Curtis, Robert Turner; <a href="/wiki/Simon_P._Norton" title="Simon P. Norton">Norton, Simon Phillips</a>; <a href="/wiki/Richard_A._Parker" title="Richard A. Parker">Parker, Richard A</a>; <a href="/wiki/Robert_Arnott_Wilson" title="Robert Arnott Wilson">Wilson, Robert Arnott</a> (1985), <i><a href="/wiki/ATLAS_of_Finite_Groups" title="ATLAS of Finite Groups">Atlas of Finite Groups</a>: Maximal Subgroups and Ordinary Characters for Simple Groups</i>, Oxford University Press, p. 85, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853199-0" title="Special:BookSources/0-19-853199-0"><bdi>0-19-853199-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Atlas+of+Finite+Groups%3A+Maximal+Subgroups+and+Ordinary+Characters+for+Simple+Groups&rft.pages=85&rft.pub=Oxford+University+Press&rft.date=1985&rft.isbn=0-19-853199-0&rft.aulast=Conway&rft.aufirst=John+Horton&rft.au=Curtis%2C+Robert+Turner&rft.au=Norton%2C+Simon+Phillips&rft.au=Parker%2C+Richard+A&rft.au=Wilson%2C+Robert+Arnott&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCarter1989" class="citation cs2"><a href="/wiki/Roger_Carter_(mathematician)" title="Roger Carter (mathematician)">Carter, Roger W.</a> (1989), <i>Simple Groups of Lie Type</i>, Wiley Classics Library, John Wiley & Sons, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-50683-4" title="Special:BookSources/0-471-50683-4"><bdi>0-471-50683-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Simple+Groups+of+Lie+Type&rft.series=Wiley+Classics+Library&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=0-471-50683-4&rft.aulast=Carter&rft.aufirst=Roger+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilson2009" class="citation cs2"><a href="/wiki/Robert_Arnott_Wilson" title="Robert Arnott Wilson">Wilson, Robert A.</a> (2009), <i>The Finite Simple Groups</i>, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 251, <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-84800-987-5" title="Special:BookSources/978-1-84800-987-5"><bdi>978-1-84800-987-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Finite+Simple+Groups&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=2009&rft.isbn=978-1-84800-987-5&rft.aulast=Wilson&rft.aufirst=Robert+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></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">Conway &al, <i>op. cit.</i>, p. 235.</span> </li> <li id="cite_note-SciAm-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-SciAm_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFA._G._LisiJ._O._Weatherall2010" class="citation journal cs1"><a href="/wiki/Antony_Garrett_Lisi" title="Antony Garrett Lisi">A. G. Lisi</a>; J. O. Weatherall (2010). <a rel="nofollow" class="external text" href="http://www.scientificamerican.com/article.cfm?id=a-geometric-theory-of-everything">"A Geometric Theory of Everything"</a>. <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i>. <b>303</b> (6): 54–61. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010SciAm.303f..54L">2010SciAm.303f..54L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fscientificamerican1210-54">10.1038/scientificamerican1210-54</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/21141358">21141358</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Scientific+American&rft.atitle=A+Geometric+Theory+of+Everything&rft.volume=303&rft.issue=6&rft.pages=54-61&rft.date=2010&rft_id=info%3Apmid%2F21141358&rft_id=info%3Adoi%2F10.1038%2Fscientificamerican1210-54&rft_id=info%3Abibcode%2F2010SciAm.303f..54L&rft.au=A.+G.+Lisi&rft.au=J.+O.+Weatherall&rft_id=http%3A%2F%2Fwww.scientificamerican.com%2Farticle.cfm%3Fid%3Da-geometric-theory-of-everything&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></span> </li> <li id="cite_note-seed-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-seed_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreg_Boustead2008" class="citation news cs1">Greg Boustead (2008-11-17). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090202122533/http://www.seedmagazine.com/news/2008/11/garrett_lisis_exceptional_appr.php">"Garrett Lisi's Exceptional Approach to Everything"</a>. <i>SEED Magazine</i>. 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=Die+Zusammensetzung+der+stetigen+endlichen+Transformationsgruppen&rft.volume=36&rft.issue=2&rft.pages=161-189&rft.date=1890&rft_id=info%3Adoi%2F10.1007%2FBF01207837&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A179178061%23id-name%3DS2CID&rft.aulast=Killing&rft.aufirst=Wilhelm&rft_id=http%3A%2F%2Fgdz.sub.uni-goettingen.de%2Findex.php%3Fid%3D11%26PPN%3DGDZPPN002252392%26L%3D1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLandsbergManivel2001" class="citation cs2">Landsberg, Joseph M.; Manivel, Laurent (2001), "The projective geometry of Freudenthal's magic square", <i><a href="/wiki/Journal_of_Algebra" title="Journal of Algebra">Journal of Algebra</a></i>, <b>239</b> (2): 477–512, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9908039">math/9908039</a></span>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjabr.2000.8697">10.1006/jabr.2000.8697</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1832903">1832903</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16320642">16320642</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Algebra&rft.atitle=The+projective+geometry+of+Freudenthal%27s+magic+square&rft.volume=239&rft.issue=2&rft.pages=477-512&rft.date=2001&rft_id=info%3Aarxiv%2Fmath%2F9908039&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1832903%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16320642%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1006%2Fjabr.2000.8697&rft.aulast=Landsberg&rft.aufirst=Joseph+M.&rft.au=Manivel%2C+Laurent&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLusztig1979" class="citation cs2"><a href="/wiki/George_Lusztig" title="George Lusztig">Lusztig, George</a> (1979), "Unipotent representations of a finite Chevalley group of type E8", <i>The Quarterly Journal of Mathematics</i>, Second Series, <b>30</b> (3): 315–338, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fqmath%2F30.3.301">10.1093/qmath/30.3.301</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/0033-5606">0033-5606</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0545068">0545068</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Quarterly+Journal+of+Mathematics&rft.atitle=Unipotent+representations+of+a+finite+Chevalley+group+of+type+E8&rft.volume=30&rft.issue=3&rft.pages=315-338&rft.date=1979&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D545068%23id-name%3DMR&rft.issn=0033-5606&rft_id=info%3Adoi%2F10.1093%2Fqmath%2F30.3.301&rft.aulast=Lusztig&rft.aufirst=George&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLusztigVogan1983" class="citation cs2"><a href="/wiki/George_Lusztig" title="George Lusztig">Lusztig, George</a>; <a href="/wiki/David_Vogan" title="David Vogan">Vogan, David</a> (1983), "Singularities of closures of K-orbits on flag manifolds", <i><a href="/wiki/Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a></i>, <b>71</b> (2), <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>: 365–379, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1983InMat..71..365L">1983InMat..71..365L</a>, <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%2FBF01389103">10.1007/BF01389103</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120917588">120917588</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=Singularities+of+closures+of+K-orbits+on+flag+manifolds&rft.volume=71&rft.issue=2&rft.pages=365-379&rft.date=1983&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120917588%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF01389103&rft_id=info%3Abibcode%2F1983InMat..71..365L&rft.aulast=Lusztig&rft.aufirst=George&rft.au=Vogan%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZamolodchikov1989" class="citation cs2"><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov, A. B.</a> (1989), "Integrals of motion and S-matrix of the (scaled) T=T<sub>c</sub> Ising model with magnetic field", <i>International Journal of Modern Physics A</i>, <b>4</b> (16): 4235–4248, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989IJMPA...4.4235Z">1989IJMPA...4.4235Z</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2FS0217751X8900176X">10.1142/S0217751X8900176X</a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1017357">1017357</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=International+Journal+of+Modern+Physics+A&rft.atitle=Integrals+of+motion+and+S-matrix+of+the+%28scaled%29+T%3DT%3Csub%3Ec%3C%2Fsub%3E+Ising+model+with+magnetic+field&rft.volume=4&rft.issue=16&rft.pages=4235-4248&rft.date=1989&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1017357%23id-name%3DMR&rft_id=info%3Adoi%2F10.1142%2FS0217751X8900176X&rft_id=info%3Abibcode%2F1989IJMPA...4.4235Z&rft.aulast=Zamolodchikov&rft.aufirst=A.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></li></ul> <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=E8_(mathematics)&action=edit&section=21" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Lusztig–Vogan_polynomial_calculation"><span id="Lusztig.E2.80.93Vogan_polynomial_calculation"></span>Lusztig–Vogan polynomial calculation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=22" title="Edit section: Lusztig–Vogan polynomial calculation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www.liegroups.org/">Atlas of Lie groups</a></li> <li><a rel="nofollow" class="external text" href="http://www.liegroups.org/kle8.html">Kazhdan–Lusztig–Vogan Polynomials for E<sub>8</sub></a></li> <li><a rel="nofollow" class="external text" href="http://atlas.math.umd.edu/kle8.narrative.html">Narrative of the Project to compute Kazhdan–Lusztig Polynomials for E<sub>8</sub></a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmerican_Institute_of_Mathematics2007" class="citation cs2"><a href="/wiki/American_Institute_of_Mathematics" title="American Institute of Mathematics">American Institute of Mathematics</a> (March 2007), <a rel="nofollow" class="external text" href="http://aimath.org/E8/"><i>Mathematicians Map E<sub>8</sub></i></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematicians+Map+E%3Csub%3E8%3C%2Fsub%3E&rft.date=2007-03&rft.au=American+Institute+of+Mathematics&rft_id=http%3A%2F%2Faimath.org%2FE8%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AE8+%28mathematics%29" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="http://golem.ph.utexas.edu/category/2007/03/news_about_e8.html">The <i>n</i>-Category Café</a>, a <a href="/wiki/University_of_Texas" class="mw-redirect" title="University of Texas">University of Texas</a> blog posting by <a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a> on E<sub>8</sub>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_links">Other links</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=E8_(mathematics)&action=edit&section=23" title="Edit section: Other links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www-math.mit.edu/~dav/e8plane.html">Graphic representation of E<sub>8</sub> root system</a>.</li> <li>The list of dimensions of <a href="/wiki/Irreducible_representation" title="Irreducible representation">irreducible representations</a> of the complex form of E<sub>8</sub> is sequence <a href="//oeis.org/A121732" class="extiw" title="oeis:A121732">A121732</a> in the <a href="/wiki/OEIS" class="mw-redirect" title="OEIS">OEIS</a>.</li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"></div><div role="navigation" class="navbox" aria-labelledby="String_theory" style="padding:3px"><table class="nowraplinks mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2" style="text-align:center;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:String_theory_topics" title="Template:String theory topics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:String_theory_topics" title="Template talk:String theory topics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:String_theory_topics" title="Special:EditPage/Template:String theory topics"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="String_theory" style="font-size:114%;margin:0 4em"><a href="/wiki/String_theory" title="String theory">String theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Background</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/String_(physics)" title="String (physics)">Strings</a></li> <li><a href="/wiki/Cosmic_string" title="Cosmic string">Cosmic strings</a></li> <li><a href="/wiki/History_of_string_theory" title="History of string theory">History of string theory</a> <ul><li><a href="/wiki/First_superstring_revolution" class="mw-redirect" title="First superstring revolution">First superstring revolution</a></li> <li><a href="/wiki/Second_superstring_revolution" class="mw-redirect" title="Second superstring revolution">Second superstring revolution</a></li></ul></li> <li><a href="/wiki/String_theory_landscape" title="String theory landscape">String theory landscape</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Nambu%E2%80%93Goto_action" title="Nambu–Goto action">Nambu–Goto action</a></li> <li><a href="/wiki/Polyakov_action" title="Polyakov action">Polyakov action</a></li> <li><a href="/wiki/Bosonic_string_theory" title="Bosonic string theory">Bosonic string theory</a></li> <li><a href="/wiki/Superstring_theory" title="Superstring theory">Superstring theory</a> <ul><li><a href="/wiki/Type_I_string_theory" title="Type I string theory">Type I string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type II string</a> <ul><li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIA string</a></li> <li><a href="/wiki/Type_II_string_theory" title="Type II string theory">Type IIB string</a></li></ul></li> <li><a href="/wiki/Heterotic_string_theory" title="Heterotic string theory">Heterotic string</a></li></ul></li> <li><a href="/wiki/N%3D2_superstring" class="mw-redirect" title="N=2 superstring">N=2 superstring</a></li> <li><a href="/wiki/F-theory" title="F-theory">F-theory</a></li> <li><a href="/wiki/String_field_theory" title="String field theory">String field theory</a></li> <li><a href="/wiki/Matrix_string_theory" title="Matrix string theory">Matrix string theory</a></li> <li><a href="/wiki/Non-critical_string_theory" title="Non-critical string theory">Non-critical string theory</a></li> <li><a href="/wiki/Non-linear_sigma_model" title="Non-linear sigma model">Non-linear sigma model</a></li> <li><a href="/wiki/Tachyon_condensation" title="Tachyon condensation">Tachyon condensation</a></li> <li><a href="/wiki/RNS_formalism" title="RNS formalism">RNS formalism</a></li> <li><a href="/wiki/GS_formalism" title="GS formalism">GS formalism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/String_duality" title="String duality">String duality</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/T-duality" title="T-duality">T-duality</a></li> <li><a href="/wiki/S-duality" title="S-duality">S-duality</a></li> <li><a href="/wiki/U-duality" title="U-duality">U-duality</a></li> <li><a href="/wiki/Montonen%E2%80%93Olive_duality" title="Montonen–Olive duality">Montonen–Olive duality</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Particles and fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Graviton" title="Graviton">Graviton</a></li> <li><a href="/wiki/Dilaton" title="Dilaton">Dilaton</a></li> <li><a href="/wiki/Tachyon" title="Tachyon">Tachyon</a></li> <li><a href="/wiki/Ramond%E2%80%93Ramond_field" title="Ramond–Ramond field">Ramond–Ramond field</a></li> <li><a href="/wiki/Kalb%E2%80%93Ramond_field" title="Kalb–Ramond field">Kalb–Ramond field</a></li> <li><a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></li> <li><a href="/wiki/Dual_graviton" title="Dual graviton">Dual graviton</a></li> <li><a href="/wiki/Dual_photon" title="Dual photon">Dual photon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Brane" title="Brane">Branes</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/D-brane" title="D-brane">D-brane</a></li> <li><a href="/wiki/NS5-brane" title="NS5-brane">NS5-brane</a></li> <li><a href="/wiki/M2-brane" title="M2-brane">M2-brane</a></li> <li><a href="/wiki/M5-brane" title="M5-brane">M5-brane</a></li> <li><a href="/wiki/S-brane" title="S-brane">S-brane</a></li> <li><a href="/wiki/Black_brane" title="Black brane">Black brane</a></li> <li><a href="/wiki/Black_hole" title="Black hole">Black holes</a></li> <li><a href="/wiki/Black_string" class="mw-redirect" title="Black string">Black string</a></li> <li><a href="/wiki/Brane_cosmology" title="Brane cosmology">Brane cosmology</a></li> <li><a href="/wiki/Quiver_diagram" title="Quiver diagram">Quiver diagram</a></li> <li><a href="/wiki/Hanany%E2%80%93Witten_transition" title="Hanany–Witten transition">Hanany–Witten transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Conformal_field_theory" title="Conformal field theory">Conformal field theory</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Virasoro_algebra" title="Virasoro algebra">Virasoro algebra</a></li> <li><a href="/wiki/Mirror_symmetry_(string_theory)" title="Mirror symmetry (string theory)">Mirror symmetry</a></li> <li><a href="/wiki/Conformal_anomaly" title="Conformal anomaly">Conformal anomaly</a></li> <li><a href="/wiki/Conformal_symmetry" title="Conformal symmetry">Conformal algebra</a></li> <li><a href="/wiki/Superconformal_algebra" title="Superconformal algebra">Superconformal algebra</a></li> <li><a href="/wiki/Vertex_operator_algebra" title="Vertex operator algebra">Vertex operator algebra</a></li> <li><a href="/wiki/Loop_algebra" title="Loop algebra">Loop algebra</a></li> <li><a href="/wiki/Kac%E2%80%93Moody_algebra" title="Kac–Moody algebra">Kac–Moody algebra</a></li> <li><a href="/wiki/Wess%E2%80%93Zumino%E2%80%93Witten_model" title="Wess–Zumino–Witten model">Wess–Zumino–Witten model</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Gauge_theory" title="Gauge theory">Gauge theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Anomaly_(physics)" title="Anomaly (physics)">Anomalies</a></li> <li><a href="/wiki/Instanton" title="Instanton">Instantons</a></li> <li><a href="/wiki/Chern%E2%80%93Simons_form" title="Chern–Simons form">Chern–Simons form</a></li> <li><a href="/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound" title="Bogomol'nyi–Prasad–Sommerfield bound">Bogomol'nyi–Prasad–Sommerfield bound</a></li> <li><a href="/wiki/Exceptional_Lie_group" class="mw-redirect" title="Exceptional Lie group">Exceptional Lie groups</a> (<a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a>, <a href="/wiki/F4_(mathematics)" title="F4 (mathematics)">F<sub>4</sub></a>, <a href="/wiki/E6_(mathematics)" title="E6 (mathematics)">E<sub>6</sub></a>, <a href="/wiki/E7_(mathematics)" title="E7 (mathematics)">E<sub>7</sub></a>, <a class="mw-selflink selflink">E<sub>8</sub></a>)</li> <li><a href="/wiki/ADE_classification" title="ADE classification">ADE classification</a></li> <li><a href="/wiki/Dirac_string" title="Dirac string">Dirac string</a></li> <li><a href="/wiki/P-form_electrodynamics" title="P-form electrodynamics"><i>p</i>-form electrodynamics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">Geometry</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Worldsheet" title="Worldsheet">Worldsheet</a></li> <li><a href="/wiki/Kaluza%E2%80%93Klein_theory" title="Kaluza–Klein theory">Kaluza–Klein theory</a></li> <li><a href="/wiki/Compactification_(physics)" title="Compactification (physics)">Compactification</a></li> <li><a href="/wiki/Why_10_dimensions" class="mw-redirect" title="Why 10 dimensions">Why 10 dimensions</a>?</li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler manifold</a></li> <li><a href="/wiki/Ricci-flat_manifold" title="Ricci-flat manifold">Ricci-flat manifold</a> <ul><li><a href="/wiki/Calabi%E2%80%93Yau_manifold" title="Calabi–Yau manifold">Calabi–Yau manifold</a></li> <li><a href="/wiki/Hyperk%C3%A4hler_manifold" title="Hyperkähler manifold">Hyperkähler manifold</a> <ul><li><a href="/wiki/K3_surface" title="K3 surface">K3 surface</a></li></ul></li> <li><a href="/wiki/G2_manifold" title="G2 manifold">G<sub>2</sub> manifold</a></li> <li><a href="/wiki/Spin(7)-manifold" title="Spin(7)-manifold">Spin(7)-manifold</a></li></ul></li> <li><a href="/wiki/Generalized_complex_structure" title="Generalized complex structure">Generalized complex manifold</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Conifold" title="Conifold">Conifold</a></li> <li><a href="/wiki/Orientifold" title="Orientifold">Orientifold</a></li> <li><a href="/wiki/Moduli_space" title="Moduli space">Moduli space</a></li> <li><a href="/wiki/Ho%C5%99ava%E2%80%93Witten_theory" title="Hořava–Witten theory">Hořava–Witten theory</a></li> <li><a href="/wiki/K-theory_(physics)" title="K-theory (physics)">K-theory (physics)</a></li> <li><a href="/wiki/Twisted_K-theory" title="Twisted K-theory">Twisted K-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Supersymmetry" title="Supersymmetry">Supersymmetry</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Supergravity" title="Supergravity">Supergravity</a></li> <li><a href="/wiki/Eleven-dimensional_supergravity" title="Eleven-dimensional supergravity">Eleven-dimensional supergravity</a></li> <li><a href="/wiki/Type_I_supergravity" title="Type I supergravity">Type I supergravity</a></li> <li><a href="/wiki/Type_IIA_supergravity" title="Type IIA supergravity">Type IIA supergravity</a></li> <li><a href="/wiki/Type_IIB_supergravity" title="Type IIB supergravity">Type IIB supergravity</a></li> <li><a href="/wiki/Superspace" title="Superspace">Superspace</a></li> <li><a href="/wiki/Lie_superalgebra" title="Lie superalgebra">Lie superalgebra</a></li> <li><a href="/wiki/Lie_supergroup" class="mw-redirect" title="Lie supergroup">Lie supergroup</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/Holography" title="Holography">Holography</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Holographic_principle" title="Holographic principle">Holographic principle</a></li> <li><a href="/wiki/AdS/CFT_correspondence" title="AdS/CFT correspondence">AdS/CFT correspondence</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%"><a href="/wiki/M-theory" title="M-theory">M-theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_theory_(physics)" title="Matrix theory (physics)">Matrix theory</a></li> <li><a href="/wiki/Introduction_to_M-theory" title="Introduction to M-theory">Introduction to M-theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="text-align:center;;width:1%">String theorists</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Mina_Aganagi%C4%87" title="Mina Aganagić">Aganagić</a></li> <li><a href="/wiki/Nima_Arkani-Hamed" title="Nima Arkani-Hamed">Arkani-Hamed</a></li> <li><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah</a></li> <li><a href="/wiki/Tom_Banks_(physicist)" title="Tom Banks (physicist)">Banks</a></li> <li><a href="/wiki/David_Berenstein" title="David Berenstein">Berenstein</a></li> <li><a href="/wiki/Raphael_Bousso" title="Raphael Bousso">Bousso</a></li> <li><a href="/wiki/Thomas_Curtright" title="Thomas Curtright">Curtright</a></li> <li><a href="/wiki/Robbert_Dijkgraaf" title="Robbert Dijkgraaf">Dijkgraaf</a></li> <li><a href="/wiki/Jacques_Distler" title="Jacques Distler">Distler</a></li> <li><a href="/wiki/Michael_R._Douglas" title="Michael R. Douglas">Douglas</a></li> <li><a href="/wiki/Michael_Duff_(physicist)" title="Michael Duff (physicist)">Duff</a></li> <li><a href="/wiki/Gia_Dvali" class="mw-redirect" title="Gia Dvali">Dvali</a></li> <li><a href="/wiki/Sergio_Ferrara" title="Sergio Ferrara">Ferrara</a></li> <li><a href="/wiki/Willy_Fischler" title="Willy Fischler">Fischler</a></li> <li><a href="/wiki/Daniel_Friedan" title="Daniel Friedan">Friedan</a></li> <li><a href="/wiki/Sylvester_James_Gates" title="Sylvester James Gates">Gates</a></li> <li><a href="/wiki/Ferdinando_Gliozzi" title="Ferdinando Gliozzi">Gliozzi</a></li> <li><a href="/wiki/Rajesh_Gopakumar" title="Rajesh Gopakumar">Gopakumar</a></li> <li><a href="/wiki/Michael_Green_(physicist)" title="Michael Green (physicist)">Green</a></li> <li><a href="/wiki/Brian_Greene" title="Brian Greene">Greene</a></li> <li><a href="/wiki/David_Gross" title="David Gross">Gross</a></li> <li><a href="/wiki/Steven_Gubser" title="Steven Gubser">Gubser</a></li> <li><a href="/wiki/Sergei_Gukov" title="Sergei Gukov">Gukov</a></li> <li><a href="/wiki/Alan_Guth" title="Alan Guth">Guth</a></li> <li><a href="/wiki/Andrew_J._Hanson" title="Andrew J. Hanson">Hanson</a></li> <li><a href="/wiki/Jeffrey_A._Harvey" title="Jeffrey A. Harvey">Harvey</a></li> <li><a href="/wiki/Gerard_%27t_Hooft" title="Gerard 't Hooft">'t Hooft</a></li> <li><a href="/wiki/Petr_Ho%C5%99ava_(theorist)" class="mw-redirect" title="Petr Hořava (theorist)">Hořava</a></li> <li><a href="/wiki/Gary_Gibbons" title="Gary Gibbons">Gibbons</a></li> <li><a href="/wiki/Shamit_Kachru" title="Shamit Kachru">Kachru</a></li> <li><a href="/wiki/Michio_Kaku" title="Michio Kaku">Kaku</a></li> <li><a href="/wiki/Renata_Kallosh" title="Renata Kallosh">Kallosh</a></li> <li><a href="/wiki/Theodor_Kaluza" title="Theodor Kaluza">Kaluza</a></li> <li><a href="/wiki/Anton_Kapustin" title="Anton Kapustin">Kapustin</a></li> <li><a href="/wiki/Igor_Klebanov" title="Igor Klebanov">Klebanov</a></li> <li><a href="/wiki/Vadim_Knizhnik" title="Vadim Knizhnik">Knizhnik</a></li> <li><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Kontsevich</a></li> <li><a href="/wiki/Oskar_Klein" title="Oskar Klein">Klein</a></li> <li><a href="/wiki/Andrei_Linde" title="Andrei Linde">Linde</a></li> <li><a href="/wiki/Juan_Mart%C3%ADn_Maldacena" class="mw-redirect" title="Juan Martín Maldacena">Maldacena</a></li> <li><a href="/wiki/Stanley_Mandelstam" title="Stanley Mandelstam">Mandelstam</a></li> <li><a href="/wiki/Donald_Marolf" title="Donald Marolf">Marolf</a></li> <li><a href="/wiki/Emil_Martinec" title="Emil Martinec">Martinec</a></li> <li><a href="/wiki/Shiraz_Minwalla" title="Shiraz Minwalla">Minwalla</a></li> <li><a href="/wiki/Greg_Moore_(physicist)" title="Greg Moore (physicist)">Moore</a></li> <li><a href="/wiki/Lubo%C5%A1_Motl" title="Luboš Motl">Motl</a></li> <li><a href="/wiki/Sunil_Mukhi" title="Sunil Mukhi">Mukhi</a></li> <li><a href="/wiki/Robert_Myers_(physicist)" title="Robert Myers (physicist)">Myers</a></li> <li><a href="/wiki/Dimitri_Nanopoulos" title="Dimitri Nanopoulos">Nanopoulos</a></li> <li><a href="/wiki/Hora%C8%9Biu_N%C4%83stase" title="Horațiu Năstase">Năstase</a></li> <li><a href="/wiki/Nikita_Nekrasov" title="Nikita Nekrasov">Nekrasov</a></li> <li><a href="/wiki/Andr%C3%A9_Neveu" title="André Neveu">Neveu</a></li> <li><a href="/wiki/Holger_Bech_Nielsen" title="Holger Bech Nielsen">Nielsen</a></li> <li><a href="/wiki/Peter_van_Nieuwenhuizen" title="Peter van Nieuwenhuizen">van Nieuwenhuizen</a></li> <li><a href="/wiki/Sergei_Novikov_(mathematician)" title="Sergei Novikov (mathematician)">Novikov</a></li> <li><a href="/wiki/David_Olive" title="David Olive">Olive</a></li> <li><a href="/wiki/Hirosi_Ooguri" title="Hirosi Ooguri">Ooguri</a></li> <li><a href="/wiki/Burt_Ovrut" title="Burt Ovrut">Ovrut</a></li> <li><a href="/wiki/Joseph_Polchinski" title="Joseph Polchinski">Polchinski</a></li> <li><a href="/wiki/Alexander_Markovich_Polyakov" title="Alexander Markovich Polyakov">Polyakov</a></li> <li><a href="/wiki/Arvind_Rajaraman" title="Arvind Rajaraman">Rajaraman</a></li> <li><a href="/wiki/Pierre_Ramond" title="Pierre Ramond">Ramond</a></li> <li><a href="/wiki/Lisa_Randall" title="Lisa Randall">Randall</a></li> <li><a href="/wiki/Seifallah_Randjbar-Daemi" title="Seifallah Randjbar-Daemi">Randjbar-Daemi</a></li> <li><a href="/wiki/Martin_Ro%C4%8Dek" title="Martin Roček">Roček</a></li> <li><a href="/wiki/Ryan_Rohm" title="Ryan Rohm">Rohm</a></li> <li><a href="/wiki/Augusto_Sagnotti" title="Augusto Sagnotti">Sagnotti</a></li> <li><a href="/wiki/Jo%C3%ABl_Scherk" title="Joël Scherk">Scherk</a></li> <li><a href="/wiki/John_Henry_Schwarz" title="John Henry Schwarz">Schwarz</a></li> <li><a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Seiberg</a></li> <li><a href="/wiki/Ashoke_Sen" title="Ashoke Sen">Sen</a></li> <li><a href="/wiki/Stephen_Shenker" title="Stephen Shenker">Shenker</a></li> <li><a href="/wiki/Warren_Siegel" title="Warren Siegel">Siegel</a></li> <li><a href="/wiki/Eva_Silverstein" title="Eva Silverstein">Silverstein</a></li> <li><a href="/wiki/%C4%90%C3%A0m_Thanh_S%C6%A1n" title="Đàm Thanh Sơn">Sơn</a></li> <li><a href="/wiki/Matthias_Staudacher" title="Matthias Staudacher">Staudacher</a></li> <li><a href="/wiki/Paul_Steinhardt" title="Paul Steinhardt">Steinhardt</a></li> <li><a href="/wiki/Andrew_Strominger" title="Andrew Strominger">Strominger</a></li> <li><a href="/wiki/Raman_Sundrum" title="Raman Sundrum">Sundrum</a></li> <li><a href="/wiki/Leonard_Susskind" title="Leonard Susskind">Susskind</a></li> <li><a href="/wiki/Paul_Townsend" title="Paul Townsend">Townsend</a></li> <li><a href="/wiki/Sandip_Trivedi" title="Sandip Trivedi">Trivedi</a></li> <li><a href="/wiki/Neil_Turok" title="Neil Turok">Turok</a></li> <li><a href="/wiki/Cumrun_Vafa" title="Cumrun Vafa">Vafa</a></li> <li><a href="/wiki/Gabriele_Veneziano" title="Gabriele Veneziano">Veneziano</a></li> <li><a href="/wiki/Erik_Verlinde" title="Erik Verlinde">Verlinde</a></li> <li><a href="/wiki/Herman_Verlinde" title="Herman Verlinde">Verlinde</a></li> <li><a href="/wiki/Julius_Wess" title="Julius Wess">Wess</a></li> <li><a href="/wiki/Edward_Witten" title="Edward Witten">Witten</a></li> <li><a href="/wiki/Shing-Tung_Yau" title="Shing-Tung Yau">Yau</a></li> <li><a href="/wiki/Tamiaki_Yoneya" title="Tamiaki Yoneya">Yoneya</a></li> <li><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Alexei_Zamolodchikov" title="Alexei Zamolodchikov">Zamolodchikov</a></li> <li><a href="/wiki/Eric_Zaslow" title="Eric Zaslow">Zaslow</a></li> <li><a href="/wiki/Bruno_Zumino" title="Bruno Zumino">Zumino</a></li> <li><a href="/wiki/Barton_Zwiebach" title="Barton Zwiebach">Zwiebach</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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