E₇ (mathematics) - Real and Complex Forms

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 is 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 is 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 E_7(-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 E_7(-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 "quateroctonionic projective plane" 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.