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<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">1.1</span> <span>Dynkin diagram</span> </div> </a> <ul id="toc-Dynkin_diagram-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Weyl/Coxeter_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Weyl/Coxeter_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Weyl/Coxeter group</span> </div> </a> <ul id="toc-Weyl/Coxeter_group-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">1.3</span> <span>Cartan matrix</span> </div> </a> <ul id="toc-Cartan_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-F4_lattice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#F4_lattice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>F<sub>4</sub> lattice</span> </div> </a> <ul id="toc-F4_lattice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Roots_of_F4" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Roots_of_F4"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>Roots of F<sub>4</sub></span> </div> </a> <ul id="toc-Roots_of_F4-sublist" class="vector-toc-list"> <li id="toc-Simple_roots" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Simple_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5.1</span> <span>Simple roots</span> </div> </a> <ul id="toc-Simple_roots-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-F4_polynomial_invariant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#F4_polynomial_invariant"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>F<sub>4</sub> polynomial invariant</span> </div> </a> <ul id="toc-F4_polynomial_invariant-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Representations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Representations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Representations</span> </div> </a> <ul id="toc-Representations-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">3</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 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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"><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 class="mw-selflink selflink">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 href="/wiki/E8_(mathematics)" title="E8 (mathematics)">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: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"><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 class="mw-selflink selflink">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 href="/wiki/E8_(mathematics)" title="E8 (mathematics)">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>F<sub>4</sub></b> is a <a href="/wiki/Lie_group" title="Lie group">Lie group</a> and also its <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <b>f</b><sub>4</sub>. It is one of the five exceptional <a href="/wiki/Simple_Lie_group" title="Simple Lie group">simple Lie groups</a>. F<sub>4</sub> has rank 4 and dimension 52. The compact form is simply connected and its <a href="/wiki/Outer_automorphism_group" title="Outer automorphism group">outer automorphism group</a> is the <a href="/wiki/Trivial_group" title="Trivial group">trivial group</a>. Its <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representation</a> is 26-dimensional. </p><p>The compact real form of F<sub>4</sub> is the <a href="/wiki/Isometry_group" title="Isometry group">isometry group</a> of a 16-dimensional <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> known as the <a href="/wiki/Octonionic_projective_plane" class="mw-redirect" title="Octonionic projective plane">octonionic projective plane</a> <b>OP</b><sup>2</sup>. 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>. </p><p>There are <a href="/wiki/List_of_simple_Lie_groups" class="mw-redirect" title="List of simple Lie groups">3 real forms</a>: a compact one, a split one, and a third one. They are the isometry groups of the three real <a href="/wiki/Albert_algebra" title="Albert algebra">Albert algebras</a>. </p><p>The F<sub>4</sub> Lie algebra may be constructed by adding 16 generators transforming as a <a href="/wiki/Spinor" title="Spinor">spinor</a> to the 36-dimensional Lie algebra <b>so</b>(9), in analogy with the construction of <a href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a>. </p><p>In older books and papers, F<sub>4</sub> is sometimes denoted by E<sub>4</sub>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Algebra">Algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=1" title="Edit section: Algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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=F4_(mathematics)&action=edit&section=2" 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 F<sub>4</sub> is: <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png" decoding="async" width="6" height="24" class="mw-file-element" data-file-width="6" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c9/Dyn-4b.png" decoding="async" width="8" height="24" class="mw-file-element" data-file-width="8" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png" decoding="async" width="6" height="24" class="mw-file-element" data-file-width="6" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Weyl/Coxeter_group"><span id="Weyl.2FCoxeter_group"></span>Weyl/Coxeter group</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=3" title="Edit section: Weyl/Coxeter group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Its <a href="/wiki/Weyl_group" title="Weyl group">Weyl</a>/<a href="/wiki/Coxeter_group" title="Coxeter group">Coxeter</a> group <span class="nowrap"><i>G</i> = <i>W</i>(F<sub>4</sub>)</span> is the <a href="/wiki/Symmetry_group" title="Symmetry group">symmetry group</a> of the <a href="/wiki/24-cell" title="24-cell">24-cell</a>: it is a <a href="/wiki/Solvable_group" title="Solvable group">solvable group</a> of order 1152. It has minimal faithful degree <span class="nowrap"><i>μ</i>(<i>G</i>) = 24</span>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> which is realized by the action on the <a href="/wiki/24-cell" title="24-cell">24-cell</a>. The group has ID (1152,157478) in the small groups library. </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=F4_(mathematics)&action=edit&section=4" title="Edit section: Cartan matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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}{rrrr}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&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 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> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>2</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> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{rrrr}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca80510babd4267c5dc6a74c5909192be8f7e030" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:22.702ex; height:12.509ex;" alt="{\displaystyle \left[{\begin{array}{rrrr}2&-1&0&0\\-1&2&-2&0\\0&-1&2&-1\\0&0&-1&2\end{array}}\right]}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="F4_lattice">F<sub>4</sub> lattice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=5" title="Edit section: F4 lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The F<sub>4</sub> <a href="/wiki/Lattice_(group)" title="Lattice (group)">lattice</a> is a four-dimensional <a href="/wiki/Body-centered_cubic" class="mw-redirect" title="Body-centered cubic">body-centered cubic</a> lattice (i.e. the union of two <a href="/wiki/Hypercubic_lattice" class="mw-redirect" title="Hypercubic lattice">hypercubic lattices</a>, each lying in the center of the other). They form a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> called the <a href="/wiki/Hurwitz_quaternion" title="Hurwitz quaternion">Hurwitz quaternion</a> ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a <a href="/wiki/24-cell" title="24-cell">24-cell</a> centered at the origin. </p> <div class="mw-heading mw-heading3"><h3 id="Roots_of_F4">Roots of F<sub>4</sub></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=6" title="Edit section: Roots of F4"><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:F4_roots_by_24-cell_duals.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/F4_roots_by_24-cell_duals.svg/220px-F4_roots_by_24-cell_duals.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/F4_roots_by_24-cell_duals.svg/330px-F4_roots_by_24-cell_duals.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/F4_roots_by_24-cell_duals.svg/440px-F4_roots_by_24-cell_duals.svg.png 2x" data-file-width="1600" data-file-height="1600" /></a><figcaption>The 24 vertices of <a href="/wiki/24-cell" title="24-cell">24-cell</a> (red) and 24 vertices of its dual (yellow) represent the 48 root vectors of F<sub>4</sub> in this <a href="/wiki/Coxeter_plane" class="mw-redirect" title="Coxeter plane">Coxeter plane</a> projection</figcaption></figure> <p>The 48 <a href="/wiki/Root_system" title="Root system">root vectors</a> of F<sub>4</sub> can be found as the vertices of the <a href="/wiki/24-cell" title="24-cell">24-cell</a> in two dual configurations, representing the vertices of a <a href="/wiki/Truncated_24-cells#Disphenoidal_288-cell" title="Truncated 24-cells">disphenoidal 288-cell</a> if the edge lengths of the 24-cells are equal: </p><p><b>24-cell vertices:</b> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c3/CDel_3.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/5/5e/CDel_node.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/8/8c/CDel_4.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/5/5e/CDel_node.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/c/c3/CDel_3.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/5/5e/CDel_node.png" decoding="async" width="5" height="23" class="mw-file-element" data-file-width="5" data-file-height="23" /></span></span></span> </p> <ul><li>24 roots by (±1, ±1, 0, 0), permuting coordinate positions</li></ul> <p><b>Dual 24-cell vertices:</b> <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/5/5e/CDel_node.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/c/c3/CDel_3.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/5/5e/CDel_node.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/8/8c/CDel_4.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/5/5e/CDel_node.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/c/c3/CDel_3.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/b/bd/CDel_node_1.png" decoding="async" width="9" height="23" class="mw-file-element" data-file-width="9" data-file-height="23" /></span></span></span> </p> <ul><li>8 roots by (±1, 0, 0, 0), permuting coordinate positions</li> <li>16 roots by (±1/2, ±1/2, ±1/2, ±1/2).</li></ul> <div class="mw-heading mw-heading4"><h4 id="Simple_roots">Simple roots</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=7" title="Edit section: Simple roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One choice of <a href="/wiki/Simple_root_(root_system)" class="mw-redirect" title="Simple root (root system)">simple roots</a> for F<sub>4</sub>, <span style="display:inline-block;"><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/b7/Dyn2-node_n1.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c8/Dyn2-3.png" decoding="async" width="6" height="24" class="mw-file-element" data-file-width="6" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/1/12/Dyn2-node_n2.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/a/a0/Dyn2-4b.png" decoding="async" width="8" height="24" class="mw-file-element" data-file-width="8" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c1/Dyn2-node_n3.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/c/c8/Dyn2-3.png" decoding="async" width="6" height="24" class="mw-file-element" data-file-width="6" data-file-height="24" /></span></span><span class="mw-default-size" typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/b/bf/Dyn2-node_n4.png" decoding="async" width="9" height="24" class="mw-file-element" data-file-width="9" data-file-height="24" /></span></span></span>, 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 {\begin{bmatrix}0&1&-1&0\\0&0&1&-1\\0&0&0&1\\{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</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>1</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>1</mn> </mtd> </mtr> <mtr> <mtd> <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> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1&-1&0\\0&0&1&-1\\0&0&0&1\\{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff06aa4793395f4e5a1eaae589fe5d204bf4dcc2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:22.877ex; height:13.509ex;" alt="{\displaystyle {\begin{bmatrix}0&1&-1&0\\0&0&1&-1\\0&0&0&1\\{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\\end{bmatrix}}}"></span></dd></dl> <p>The Hasse diagram for the F<sub>4</sub> root poset is shown below right. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:F4HassePoset.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/F4HassePoset.svg/300px-F4HassePoset.svg.png" decoding="async" width="300" height="504" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6e/F4HassePoset.svg/450px-F4HassePoset.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6e/F4HassePoset.svg/600px-F4HassePoset.svg.png 2x" data-file-width="890" data-file-height="1494" /></a><figcaption><a href="/wiki/Hasse_diagram" title="Hasse diagram">Hasse diagram</a> of F<sub>4</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> <div class="mw-heading mw-heading3"><h3 id="F4_polynomial_invariant">F<sub>4</sub> polynomial invariant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=8" title="Edit section: F4 polynomial invariant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Just as O(<i>n</i>) is the group of automorphisms which keep the quadratic polynomials <span class="nowrap"><i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> + ...</span> invariant, F<sub>4</sub> is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables). </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 C_{1}=x+y+z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}=x+y+z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae0fd9d72cccf27554f2137718c164fc7d9aeb5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.069ex; height:2.509ex;" alt="{\displaystyle C_{1}=x+y+z}"></span></dd> <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 C_{2}=x^{2}+y^{2}+z^{2}+2X{\overline {X}}+2Y{\overline {Y}}+2Z{\overline {Z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mn>2</mn> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mn>2</mn> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{2}=x^{2}+y^{2}+z^{2}+2X{\overline {X}}+2Y{\overline {Y}}+2Z{\overline {Z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46601e479de89afe40d7ff60ea2c7004740bfc05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:41.831ex; height:3.343ex;" alt="{\displaystyle C_{2}=x^{2}+y^{2}+z^{2}+2X{\overline {X}}+2Y{\overline {Y}}+2Z{\overline {Z}}}"></span></dd> <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 C_{3}=xyz-xX{\overline {X}}-yY{\overline {Y}}-zZ{\overline {Z}}+XYZ+{\overline {XYZ}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mi>x</mi> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>y</mi> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>−</mo> <mi>z</mi> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mi>X</mi> <mi>Y</mi> <mi>Z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>X</mi> <mi>Y</mi> <mi>Z</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{3}=xyz-xX{\overline {X}}-yY{\overline {Y}}-zZ{\overline {Z}}+XYZ+{\overline {XYZ}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5786fc1e64838bac165ba931a996915032a60e0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:49.79ex; height:3.343ex;" alt="{\displaystyle C_{3}=xyz-xX{\overline {X}}-yY{\overline {Y}}-zZ{\overline {Z}}+XYZ+{\overline {XYZ}}}"></span></dd></dl> <p>Where <i>x</i>, <i>y</i>, <i>z</i> are real-valued and <i>X</i>, <i>Y</i>, <i>Z</i> are octonion valued. Another way of writing these invariants is as (combinations of) Tr(<i>M</i>), Tr(<i>M</i><sup>2</sup>) and Tr(<i>M</i><sup>3</sup>) of the <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">hermitian</a> <a href="/wiki/Octonion" title="Octonion">octonion</a> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</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 M={\begin{bmatrix}x&{\overline {Z}}&Y\\Z&y&{\overline {X}}\\{\overline {Y}}&X&z\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Z</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi>Y</mi> </mtd> </mtr> <mtr> <mtd> <mi>Z</mi> </mtd> <mtd> <mi>y</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>X</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Y</mi> <mo accent="false">¯</mo> </mover> </mrow> </mtd> <mtd> <mi>X</mi> </mtd> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\begin{bmatrix}x&{\overline {Z}}&Y\\Z&y&{\overline {X}}\\{\overline {Y}}&X&z\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04df77fb9e8f9a22d5455bf5e27d776435744998" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:20.312ex; height:10.843ex;" alt="{\displaystyle M={\begin{bmatrix}x&{\overline {Z}}&Y\\Z&y&{\overline {X}}\\{\overline {Y}}&X&z\end{bmatrix}}}"></span></dd></dl> <p>The set of polynomials defines a 24-dimensional compact surface. </p> <div class="mw-heading mw-heading2"><h2 id="Representations">Representations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=9" title="Edit section: Representations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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/A121738" class="extiw" title="oeis:A121738">A121738</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, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749, 76076, 81081, 100776, 106496, 107406, 119119, 160056 (twice), 184756, 205751, 212992, 226746, 340119, 342056, 379848, 412776, 420147, 627912...</dd></dl> <p>The 52-dimensional representation is the <a href="/wiki/Adjoint_representation_of_a_Lie_algebra" class="mw-redirect" title="Adjoint representation of a Lie algebra">adjoint representation</a>, and the 26-dimensional one is the trace-free part of the action of F<sub>4</sub> on the exceptional <a href="/wiki/Albert_algebra" title="Albert algebra">Albert algebra</a> of dimension 27. </p><p>There are two non-isomorphic irreducible representations of dimensions 1053, 160056, 4313088, etc. The <a href="/wiki/Fundamental_representation" title="Fundamental representation">fundamental representations</a> are those with dimensions 52, 1274, 273, 26 (corresponding to the four nodes in the <a href="#Dynkin_diagram">Dynkin diagram</a> in the order such that the double arrow points from the second to the third). </p><p>Embeddings of the maximal subgroups of F<sub>4</sub> up to dimension 273 with associated projection matrix are shown below. </p><p><span typeof="mw:File"><a href="/wiki/File:F4_Maximal_Embeddings.svg" class="mw-file-description" title="Embeddings of the maximal subgroups of F4 up to dimension 273 with associated projection matrix."><img alt="Embeddings of the maximal subgroups of F4 up to dimension 273 with associated projection matrix." src="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/F4_Maximal_Embeddings.svg/550px-F4_Maximal_Embeddings.svg.png" decoding="async" width="550" height="473" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/07/F4_Maximal_Embeddings.svg/825px-F4_Maximal_Embeddings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/07/F4_Maximal_Embeddings.svg/1100px-F4_Maximal_Embeddings.svg.png 2x" data-file-width="549" data-file-height="472" /></a></span> </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=F4_(mathematics)&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/24-cell" title="24-cell">24-cell</a></li> <li><a href="/wiki/Albert_algebra" title="Albert algebra">Albert algebra</a></li> <li><a href="/wiki/Cayley_plane" title="Cayley plane">Cayley plane</a></li> <li><a href="/wiki/Dynkin_diagram" title="Dynkin diagram">Dynkin diagram</a></li> <li><a href="/wiki/Fundamental_representation" title="Fundamental representation">Fundamental representation</a></li> <li><a href="/wiki/Simple_Lie_group" title="Simple Lie group">Simple Lie group</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=F4_(mathematics)&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><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="CITEREFSaunders2014" class="citation arxiv cs1">Saunders, Neil (2014). "Minimal Faithful Permutation Degrees for Irreducible Coxeter Groups and Binary Polyhedral Groups". <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/0812.0182">0812.0182</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.GR">math.GR</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=Minimal+Faithful+Permutation+Degrees+for+Irreducible+Coxeter+Groups+and+Binary+Polyhedral+Groups&rft.date=2014&rft_id=info%3Aarxiv%2F0812.0182&rft.aulast=Saunders&rft.aufirst=Neil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AF4+%28mathematics%29" class="Z3988"></span></span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAdams1996" class="citation book cs1">Adams, J. Frank (1996). <a rel="nofollow" class="external text" href="https://books.google.com/books?isbn=0226005275"><i>Lectures on exceptional Lie groups</i></a>. Chicago Lectures in Mathematics. <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-226-00526-3" title="Special:BookSources/978-0-226-00526-3"><bdi>978-0-226-00526-3</bdi></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=1428422">1428422</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+exceptional+Lie+groups&rft.series=Chicago+Lectures+in+Mathematics&rft.pub=University+of+Chicago+Press&rft.date=1996&rft.isbn=978-0-226-00526-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1428422%23id-name%3DMR&rft.aulast=Adams&rft.aufirst=J.+Frank&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fisbn%3D0226005275&rfr_id=info%3Asid%2Fen.wikipedia.org%3AF4+%28mathematics%29" class="Z3988"></span></li> <li><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>, <i>The Octonions</i>, Section 4.2: F<sub>4</sub>, <a rel="nofollow" class="external text" href="https://www.ams.org/bull/2002-39-02/S0273-0979-01-00934-X/home.html">Bull. Amer. Math. Soc. <b>39</b> (2002), 145-205</a>. Online HTML version at <a rel="nofollow" class="external free" href="http://math.ucr.edu/home/baez/octonions/node15.html">http://math.ucr.edu/home/baez/octonions/node15.html</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChevalleySchafer1950" class="citation journal cs1">Chevalley C, Schafer RD (February 1950). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063148">"The Exceptional Simple Lie Algebras F(4) and E(6)"</a>. <i>Proc. Natl. Acad. Sci. U.S.A</i>. <b>36</b> (2): 137–41. <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/1950PNAS...36..137C">1950PNAS...36..137C</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.36.2.137">10.1073/pnas.36.2.137</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063148">1063148</a></span>. <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/16588959">16588959</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc.+Natl.+Acad.+Sci.+U.S.A.&rft.atitle=The+Exceptional+Simple+Lie+Algebras+F%284%29+and+E%286%29&rft.volume=36&rft.issue=2&rft.pages=137-41&rft.date=1950-02&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063148%23id-name%3DPMC&rft_id=info%3Apmid%2F16588959&rft_id=info%3Adoi%2F10.1073%2Fpnas.36.2.137&rft_id=info%3Abibcode%2F1950PNAS...36..137C&rft.aulast=Chevalley&rft.aufirst=C&rft.au=Schafer%2C+RD&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063148&rfr_id=info%3Asid%2Fen.wikipedia.org%3AF4+%28mathematics%29" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson1971" class="citation book cs1"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (1971-06-01). <i>Exceptional Lie Algebras</i> (1st ed.). CRC Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8247-1326-5" title="Special:BookSources/0-8247-1326-5"><bdi>0-8247-1326-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Exceptional+Lie+Algebras&rft.edition=1st&rft.pub=CRC+Press&rft.date=1971-06-01&rft.isbn=0-8247-1326-5&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AF4+%28mathematics%29" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output 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.navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Exceptional_Lie_groups" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><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:Exceptional_Lie_groups" title="Template:Exceptional Lie groups"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Exceptional_Lie_groups" title="Template talk:Exceptional Lie groups"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Exceptional_Lie_groups" title="Special:EditPage/Template:Exceptional Lie groups"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Exceptional_Lie_groups" style="font-size:114%;margin:0 4em"><a href="/wiki/Simple_Lie_group#Exceptional_cases" title="Simple Lie group">Exceptional Lie groups</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/G2_(mathematics)" title="G2 (mathematics)">G<sub>2</sub></a></li> <li><a class="mw-selflink selflink">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 href="/wiki/E8_(mathematics)" title="E8 (mathematics)">E<sub>8</sub></a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" 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 class="mw-selflink selflink">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 href="/wiki/E8_(mathematics)" title="E8 (mathematics)">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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