Isomorphisms of Chevalley groups over algebras
The aim of this paper is to characterize the automorphisms and isomorphisms of Chevalley groups over \(\mathbb{Q}\)-algebras without zero divisors. It is based on an earlier work of the author [Tôhoku Math. J., II. Ser. 47, No. 1, 81-97 (1995; Zbl 0829.20070)]. Let \(G\) be an almost simple Chevalley-Demazure group scheme of rank greater than 1, and \(R\) a commutative \(\mathbb{Q}\)-algebra without zero divisors. The author proves that any automorphism of \(G(R)\) is ``standard'' in some sense. Let \(G\) and \(G'\) be Chevalley-Demazure group schemes containing no simple component of type \(A_1\), and let \(R\) and \(R'\) be commutative \(\mathbb{Q}\)-algebras without zero divisors. The author proves that, if \(G\) is simply connected, then any isomorphism from \(G(R)\) onto \(G'(R')\) is ``standard''. The problem to determine the isomorphisms of Chevalley groups of type \(A_1\) is more complicated and is still open.
almost simple Chevalley-Demazure group schemes
scientific article; zbMATH DE number 1448608
scientific article