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Lie theory

Exceptional Lie algebras

In 1893, Élie Cartan published a note describing an open set in {$\mathbb {C} ^{5}$} equipped with a 2-dimensional distribution—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra {${\mathfrak{g}}_2$} appears as the infinitesimal symmetries. In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.

In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of {$G_2$}.

In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the compact real form of {$G_2$}.

Note: α and A together form root system identical to {$A_2$} while the system formed by β and B is isomorphic to {$A_2$}.

Its Weyl/Coxeter group {$G=W(G_{2})$} is the dihedral group D_6 of order 12.

Polygon is 1 6 6 2 oriented faces du tetrahedrai dualumas.