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In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 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 E7 algebra is thus one of the five exceptional cases. The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56. ==Real and complex forms== There is a unique complex Lie algebra of type E7, corresponding to a complex group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension 266. This has fundamental group Z/2Z, has maximal compact subgroup the compact form (see below) of E7, and has an outer automorphism group of order 2 generated by complex conjugation. As well as the complex Lie group of type E7, there are four real forms of the Lie algebra, and correspondingly four 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 133, as follows: * The compact form (which is usually the one meant if no other information is given), which has fundamental group Z/2Z and has trivial outer automorphism group. * The split form, EV (or E7(7)), which has maximal compact subgroup SU(8)/, fundamental group cyclic of order 4 and outer automorphism group of order 2. * EVI (or E7(-5)), which has maximal compact subgroup SU(2)·SO(12)/(center), fundamental group non-cyclic of order 4 and trivial outer automorphism group. * EVII (or E7(-25)), which has maximal compact subgroup SO(2)·E6/(center), infinite cyclic findamental group and outer automorphism group of order 2. For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups. The compact real form of E7 is the isometry group of the 64-dimensional exceptional compact Riemannian symmetric space EVI (in Cartan's classification). It is known informally as the "" because it can be built using an algebra that is the tensor product of the quaternions and the octonions, and is also known as a Rosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the ''magic square'', due to Hans Freudenthal and Jacques Tits. The Tits–Koecher construction produces forms of the E7 Lie algebra from Albert algebras, 27-dimensional exceptional Jordan algebras. 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「E7 (mathematics)」の詳細全文を読む スポンサード リンク
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