Automorphisms and Isomorphisms of Chevalley Groups and Algebras

Suppose that Φ is a reduced irreducible root system, R is an associative commutative ring with unity, G(Φ, R) is the corresponding adjoint Chevalley group, and E(Φ, R) is its elementary subgroup (see Section 5). There are a lot of results (see, e.g., [Wat80], [Pet82], [GMi83], [HO’M89], [Abe93], [Che00], [Bun07], and references therein*) asserting that, under some conditions, all automorphisms of Chevalley (or similar) groups are standard in some sense (depending on what a particular author succeeded to prove). In this paper, we use the most universal and natural definition of standardness suggested by A. E. Zalesskii [Zal83]: an automorphism of an adjoint Chevalley group is called standard if it is induced by an automorphism of the corresponding Lie algebra. More precisely, this means the following. Clearly, E(Φ, R) and G(Φ, R) embed naturally into the automorphism group of the corresponding Lie algebra L(Φ, R) over R. A slightly less obvious fact is that (under some conditions, see Section 5) both groups are normal in AutRL(Φ, R) and even in the larger group Aut ZL(Φ, R) = Aut ZR⋌AutRL(Φ, R) consisting of the automorphisms of this algebra considered as a Lie ring. Thus, each automorphism f ∈ Aut ZL(Φ, R) of the Lie ring induces an automorphism f : g 7→ fgf of the Chevalley groups G(Φ, R) and E(Φ, R). The main results of this paper are the following theorems.