Automorphisms of Chevalley groups over commutative rings

Abstract

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank > 1 over a commutative ring (with 1/2 for the systems ${\mathbf{A}}_2,{\mathbf{F}}_4,{\mathbf{B}}_l,{\mathbf{C}}_l$; with 1/2 and 1/3 for the system ${\mathbf{G}}_2$) is standard, i.e., it is a composition of ring, inner, central and graph automorphisms. This result finalizes description of automorphisms of Chevalley groups. However, the restrictions on invertible elements can be a topic of further considerations. We provide also some model-theoretic applications of this description.

KEYWORDS:

• Automorphisms
• Chevalley groups
• commutative rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20G35

Acknowledgments

My sincere thanks go to Eugene Plotkin for very useful discussions regarding various aspects of this work and permanent attention to it.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.