Automorphisms of a Chevalley group of type \(G_2\) over a commutative ring \(R\) with \(1/3\) generated by the invertible elements and \(2R\)
The study of automorphisms of a classical group has a rich history. One milestone result of this line of research is that of \textit{R. Steinberg} [Can. J. Math. 12, 606--615 (1960; Zbl 0097.01703)], which states that the automorphisms of a Chevalley group \(G\) over a finite field are a composition of four types of automorphisms, namely (1) inner automorphisms; (2) diagonal automorphisms; (3) field automorphisms; and (4) graph automorphisms (related to Dynkin diagram of the associated Lie algebra). Any automorphism which is a composition of the above four types (if they exist) is usually referred to as \textit{standard} automorphism in the literature. The scenario for automorphisms of classical groups defined over a general ring is quite different. For example, the automorphisms of the group \(\mathrm{GL}_3(R)\) are not always standard.\N\NThe first author proved many useful results in this direction, for example she proved that every automorphism of an elementary adjoint Chevalley group of type \(A_l\), \(B_l\) or \(E_l\) over a local commutative ring with \(1/2\) is a composition of a ring automorphism and conjugation by some matrix from the normalizer of that Chevalley group in \(\mathrm{GL}(V )\) (here \(V\) is an adjoint representation space) and hence of standard type [the first author, Algebra Logic 48, No. 4, 250--267 (2009; Zbl 1245.20063); translation from Algebra Logika 48, No. 4, 443--470 (2009)]. \N\NThere are similar results about the standardity of automorphisms of type \(G_2\) for arbitrary commutative rings, for example, every automorphism of an (elementary) adjoint Chevalley group with root system \(G_2\) over a commutative ring with \(1/2\) and \(1/3\) is standard, see [the first author, J. Algebra 355, No. 1, 154--170 (2012; Zbl 1260.20071)]. The present article explores the automorphism of \(G_2\) over a ring \(R\) which is a commutative ring with \(1/3\) and is generated by its invertible elements and the ideal \(2R\).\N\NThe authors prove the following: Let \(R\) be a commutative ring with \(1/3 \), generated by its invertible elements and the ideal \(2R\), and \(G = G_{\mathrm{ad}}(G_2, R)\) or \(G = E_{\mathrm{ad}}(G_2, R)\) be a Chevalley group of type \(G_2\) or its elementary subgroup. Then, any automorphism of group \(G\) is standard, i.e., a composition of a ring automorphism (i.e. the automorphism \((a_{ij})\mapsto(\rho(a_{ij}))\) associated with a ring automorphism \(\rho:R\longrightarrow R\)) and strictly inner automorphisms (i.e. the conjugating element \(g\) satisfies \(g\in G_{\mathrm{ad}}(G_2, R)\)).
scientific article; zbMATH DE number 7915933
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