E₇ (mathematics) - E₇ As An Algebraic Group

E₇ As An Algebraic Group

By means of a Chevalley basis for the Lie algebra, one can define E₇ as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) adjoint form of E₇. Over an algebraically closed field, this and its double cover are the only forms; however, over other fields, there are often many other forms, or “twists” of E₇, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k, Aut(E₇)) which, because the Dynkin diagram of E₇ (see below) has no automorphisms, coincides with H1(k, E_{7, ad}).

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E₇ coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E₇ have fundamental group Z/2Z in the sense of algebraic geometry, meaning that they admit exactly one double cover; the further non-compact real Lie group forms of E₇ are therefore not algebraic and admit no faithful finite-dimensional representations.

Over finite fields, the Lang–Steinberg theorem implies that H1(k, E₇) = 0, meaning that E₇ has no twisted forms: see below.