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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 ). This classifies Lie algebras 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 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. ==Real and complex forms== There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see below) of E6, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism. As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows: *The compact form (which is usually the one meant if no other information is given), which has fundamental group Z/3Z and outer automorphism group Z/2Z. *The split form, EI (or E6(6)), which has maximal compact subgroup Sp(4)/(±1), fundamental group of order 2 and outer automorphism group of order 2. *The quasi-split form EII (or E6(2)), which has maximal compact subgroup SU(2) × SU(6)/(center), fundamental group cyclic of order 6 and outer automorphism group of order 2. *EIII (or E6(-14)), which has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group Z and trivial outer automorphism group. *EIV (or E6(-26)), which has maximal compact subgroup F4, trivial fundamental group cyclic and outer automorphism group of order 2. The EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective plane OP2.〔 (theorem 7.4 on page 335, and following paragraph).〕 It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. The compact real form of E6 is the isometry group of a 32-dimensional Riemannian manifold known as the 'bioctonionic projective plane'; similar constructions for E7 and E8 are known as the Rosenfeld projective planes, and are part of the Freudenthal magic square. 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「E6 (mathematics)」の詳細全文を読む スポンサード リンク
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