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In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras , as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14. The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation. == History == The Lie algebra , being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23, 1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now call . In 1893, Élie Cartan published a note describing an open set in 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 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 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 G2. 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 G2. In older books and papers, G2 is sometimes denoted by E2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「G2 (mathematics)」の詳細全文を読む スポンサード リンク
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