Automorphisms of a Chevalley group of type \(G_2\) over a commutative ring \(R\) with \(1/3\) generated by the invertible elements and \(2R\)
type: Item
instance of: scholarly article
title: Automorphisms of a Chevalley group of type \(G_2\) over a commutative ring \(R\) with \(1/3\) generated by the invertible elements and \(2R\)
author: E. I. Bunina
author: M. A. Vladykina
published in: Journal of Mathematical Sciences (New York)
publication date: 2024-09-19T00:00:00Z
review text: 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…
reviewed by: Saikat Panja
Mathematics Subject Classification ID: 20H25
Mathematics Subject Classification ID: 20E36
zbMATH DE Number: 7915933
zbMATH Keywords: automorphisms
zbMATH Keywords: Chevalley groups over rings
MaRDI profile type: Publication
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DOI: 10.1007/S10958-024-07361-0
description: scientific article; zbMATH DE number 7915933