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Concept# E7 (mathematics)

Summary

DISPLAYTITLE:E7 (mathematics)
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 An, Bn, Cn, Dn, 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.
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)/{±1}, 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.

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