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 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.

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.

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