Steinberg Groups
Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-split orthogonal groups. Steinberg (1959) found a modification of Chevalley's construction that gave these groups and two new families 3D4, 2E6, the second of which was discovered at about the same time from a different point of view by Tits (1958). This construction generalizes the usual construction of the unitary group from the general linear group.
The unitary group arises as follows: the general linear group over the complex numbers has a diagram automorphism given by reversing the Dynkin diagram An (which corresponds to taking the transpose inverse), and a field automorphism given by taking complex conjugation, which commute. The unitary group is the group of fixed points of the product of these two automorphisms.
In the same way, many Chevalley groups have diagram automorphisms induced by automorphisms of their Dynkin diagrams, and field automorphisms induced by automorphisms of a finite field. Analogously to the unitary case, Steinberg constructed families of groups by taking fixed points of a product of a diagram and a field automorphism.
These gave:
- the unitary groups 2An, from the order 2 automorphism of An;
- further orthogonal groups 2Dn, from the order 2 automorphism of Dn;
- the new series 2E6, from the order 2 automorphism of E6;
- the new series 3D4, from the order 3 automorphism of D4.
The groups of type 3D4 have no analogue over the reals, as the complex numbers have no automorphism of order 3. The symmetries of the D4 diagram also give rise to triality.
Read more about this topic: Group Of Lie Type
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