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In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled A''n'', B''n'', C''n'', D''n'', and five exceptional cases labeled E6, E7, E8, F4, and G2. The E8 algebra is the largest and most complicated of these exceptional cases. discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by Élie Cartan. Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them gives rise to a simple Lie group of dimension 248, exactly one of which is compact. introduced algebraic groups and Lie algebras of type E8 over other fields: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type. == Basic description == The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is 8. Therefore the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 21435527 = 696729600. The compact group E8 is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself; it is also the unique one which has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length). There is a Lie algebra E''n'' for every integer ''n'' ≥ 3, which is infinite dimensional if ''n'' is greater than 8. 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「E8 (mathematics)」の詳細全文を読む スポンサード リンク
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