Construction of Lie Groups Via Diagram Automorphisms
Diagram automorphisms in turn yield additional Lie groups and groups of Lie type, which are of central importance in the classification of finite simple groups.
The Chevalley group construction of Lie groups in terms of their Dynkin diagram does not yield some of the classical groups, namely the unitary groups and the non-split orthogonal groups. The Steinberg groups construct the unitary groups 2An, while the other orthogonal groups are constructed as 2Dn, where in both cases this refers to combining a diagram automorphism with a field automorphism. This also yields additional exotic Lie groups 2E6 and 3D4, the latter only defined over fields with an order 3 automorphism.
The additional diagram automorphisms in positive characteristic yield the Suzuki–Ree groups, 2B2, 2F4, and 2G2.
Read more about this topic: Dynkin Diagram, Automorphisms
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