E₆ (mathematics) - E₆ As An Algebraic Group

E₆ As An Algebraic Group

By means of a Chevalley basis for the Lie algebra, one can define E₆ 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 E₆. Over an algebraically closed field, this and its triple cover are the only forms; however, over other fields, there are often many other forms, or “twists” of E₆, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k, Aut(E₆)) which, because the Dynkin diagram of E₆ (see below) has automorphism group Z/2Z, maps to H1(k, Z/2Z) = Hom (Gal(k), Z/2Z) with kernel H1(k, E_6,ad).

Over the field of real numbers, the real component of the identity of these algebraically twisted forms of E₆ coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all adjoint forms of E₆ have fundamental group Z/3Z in the sense of algebraic geometry, with Galois action as on the third roots of unity; this means that they admit exactly one triple cover (which may be trivial on the real points); the further non-compact real Lie group forms of E₆ are therefore not algebraic and admit no faithful finite-dimensional representations. The compact real form of E₆ as well as the noncompact forms EI=E₆₍₆₎ and EIV=E_6(-26) are said to be inner or of type 1E₆ meaning that their class lies in H1(k, E_6,ad) or that complex conjugation induces the trivial automorphism on the Dynkin diagram, whereas the other two real forms are said to be outer or of type 2E₆.

Over finite fields, the Lang–Steinberg theorem implies that H1(k, E₆) = 0, meaning that E₆ has exactly one twisted form, known as 2E₆: see below.