Automorphisms of elementary adjoint Chevalley groups of types A l , D l , and E l over local rings with 1/2

Abstract

In this paper we prove that every automorphism of a (elementary) Chevalley group of type A l , D l , or E l , l 2, over a commutative local ring with 1/2 is standard, i. e., is the composition of inner, ring, graph and central automorphisms.

Automorphisms of Chevalley groups of different types over commutative rings

“…In the paper [15] of the author it was shown that automorphisms of adjoint elementary Chevalley groups with root systems A l , D l , E l , l 2, over local rings with invertible 2 can be represented as the composition of ring automorphism and an automorphism-conjugation, where as automorphism-conjugation we call a conjugation of elements of a Chevalley group in the adjoint representation by some matrix from the normalizer of this group in GL (V ). In the paper [17] according to the results of [15] it was proved that every automorphism of an arbitrary (elementary) Chevalley group of the described type is standard, i. e., it is represented as the composition of ring, inner, central and graph automorphism. In the same paper it was obtained the theorem describing the normalizer of Chevalley groups in their adjoint representation, which also holds for local rings without 1/2.…”

“…In the present paper with the help of results of author's papers [15], [17], [19], [16], [18], [20], and also the methods, described by V.M. Petechuk in [51] for the special linear group SL , we describe automorphisms of adjoint Chevalley groups over arbitrary commutative rings with the assumption that the corresponding root systems have rank > 1, for the root systems A 2 , F 4 , B l , C l the ring contains 1/2, for the system G 2 the ring contains 1/2 and 1/3.…”

Automorphisms of Chevalley groups of type F4 over local rings with 1/2

“…Since in the papers [5] and [6] the root system in there second sections was arbitrary, we can suppose all results of these sections to be proved also for our root system F 4 .…”

“…6) the position (51, 5) gives z 41 = 0, the position (51, 6) gives z 40 = 0, the position (52, 7) gives z 39 = 0, the position (51, 8) gives z 10 = 0, the position (52, 8) gives z 38 = 1.…”

“…Its position (13,14) gives t 5 = 0, the position (5,5) gives t 4 = 1/t 2 2 , (6, 6) gives t 7 = t 2 2 ; (3, 19) gives t 9 = 0; (19,19) gives t 10 = 1/t 8 .…”