𝐸₈, the most exceptional group

2016/05/04 by Skip Garibaldi, S. Garibaldi · 2 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic group #Algebraic number #Algebraic structures and combinatorial models #Combinatorics #Epistemology #Geometry #Group (periodic table) #Lie algebra #Lie group #Mathematical analysis #Mathematics #Philosophy #Physics #Point (geometry) #Pure mathematics #Quantum mechanics #Simple (philosophy) #math.GR #math.RA #math.RT #msc:17B25 #msc:20G15 #msc:20G41

paper · pdf · doi:10.1090/bull/1540

published as Bulletin of the AMS, vol. 53 #4 (2016), 643--671

arxiv created 2016/05/04 · openalex publication_date 2016/06/29 · arxiv updated 2016/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

The five exceptional simple Lie algebras over the complex number are included one within the other as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German g 2 subset-of German f 4 subset-of German e 6 subset-of German e 7 subset-of German e 8"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">g</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">f</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">e</mml:mi> </mml:mrow> <mml:mn>6</mml:mn> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">e</mml:mi> </mml:mrow> <mml:mn>7</mml:mn> </mml:msub> <mml:mo> ⊂ </mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">e</mml:mi> </mml:mrow> <mml:mn>8</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak g₂ ⊂ \mathfrak f₄ ⊂ \mathfrak e₆ ⊂ \mathfrak e₇ ⊂ \mathfrak e₈</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The biggest one, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German e 8"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">e</mml:mi> </mml:mrow> <mml:mn>8</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathfrak e₈</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is in many ways the most mysterious. This article surveys what is known about it, including many recent results, and it focuses on the point of view of Lie algebras and algebraic groups over fields.

Citations

• Introduction to Affine Group Schemes
• Sporadic and Exceptional
• Representation Theory
• On conjugacy classes in the Lie group E₈
• Problems of classifying associative or Lie algebras and triples of symmetric or skew-symmetric matrices are wild
• INTEGRALS OF MOTION AND S-MATRIX OF THE (SCALED) T = Tc ISING MODEL WITH MAGNETIC FIELD
• Restricting the Rost invariant to the center
• Elementary abelian p-subgroups of algebraic groups
• Finite-Dimensional Division Algebras over Fields
• Finite simple groups which projectively embed in an exceptional Lie group are classified!
• There is No “Theory of Everything” Inside E8
• The greatest mathematical paper of all time
• The Kneser-Tits conjecture for groups with Tits-index E_8,2⁶⁶ over an arbitrary field
• The algebra of grand unified theories
• Emergence of the Theory of Lie Groups
• Die Zusammensetzung der stetigen endlichen Transformationsgruppen

Cited by

• The bracket of the exceptional Lie algebra E8
• E8 (mathematics) [wikipedia]

Related

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• Sporadic SICs and Exceptional Lie Algebras
• Gradings on o(8,\mathbb C)
• On realizations of the Lie groups G_2,\boldmath\scriptstyleH,F_4,\boldmath\scriptstyleH,E_6,\boldmath\scriptstyleH,E_7,\boldmath\scriptstyleH,E_8,\boldmath\scriptstyleH , second edition
• Linear models of the exceptional Lie algebra \mathfrake₈