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What (who) is E6 (mathematics) - definition
![root poset]] with edge labels identifying added simple root position](https://commons.wikimedia.org/wiki/Special:FilePath/E6HassePoset.svg?width=200 "root poset]] with edge labels identifying added simple root position")
![[[Omar Khayyám]]'s "Cubic equations and intersections of conic sections" the first page of the two-chaptered manuscript kept in Tehran University](https://commons.wikimedia.org/wiki/Special:FilePath/Khayyam-paper-1stpage.png?width=200 "[[Omar Khayyám]]'s \"Cubic equations and intersections of conic sections\" the first page of the two-chaptered manuscript kept in Tehran University")
![To solve the third-degree equation ''x''<sup>3</sup> + ''a''<sup>2</sup>''x'' = ''b'' Khayyám constructed the [[parabola]] ''x''<sup>2</sup> = ''ay'', a [[circle]] with diameter ''b''/''a''<sup>2</sup>, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the ''x''-axis.](https://commons.wikimedia.org/wiki/Special:FilePath/Omar Kayyám - Geometric solution to cubic equation.svg?width=200 "To solve the third-degree equation ''x''<sup>3</sup> + ''a''<sup>2</sup>''x'' = ''b'' Khayyám constructed the [[parabola]] ''x''<sup>2</sup> = ''ay'', a [[circle]] with diameter ''b''/''a''<sup>2</sup>, and a vertical line through the intersection point. The solution is given by the length of the horizontal line segment from the origin to the intersection of the vertical line and the ''x''-axis.")
![Engraving of [[Abū Sahl al-Qūhī]]'s perfect compass to draw conic sections.](https://commons.wikimedia.org/wiki/Special:FilePath/Gravure originale du compas parfait par Abū Sahl al-Qūhī.jpg?width=200 "Engraving of [[Abū Sahl al-Qūhī]]'s perfect compass to draw conic sections.")
![theorem of Ibn Haytham]].](https://commons.wikimedia.org/wiki/Special:FilePath/Theorem of al-Haitham.JPG?width=200 "theorem of Ibn Haytham]].")
Wikipedia
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.
The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional (complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27-dimensional.
In particle physics, E6 plays a role in some grand unified theories.
