Automorphisms of Chevalley groups of types B 2 and G 2 over local rings

Abstract

In the paper we prove that every automorphism of any adjoint Chevalley group of types B 2 or G 2 is standard, i.e., it is a composition of the "inner" automorphism, ring automorphism and central automorphism.Introduction.An associative ring R with a unit is called local, if it has exactly one maximal ideal (which coincides with the Jacobson radical of R). Equivalently, all non-invertible elements of R form an ideal. In this paper all rings under consideration are commutative.Let G ad be a Chevalley-Demazure group scheme associated with an irreducible root system Φ of type B 2 or G 2 (see detailed definitions in the next section); G ad (R, Φ) be a set of points G with values in R; E ad (R, Φ) be the elementary subgroup of G ad (R, Φ), where R is a local commutative ring with 1. In this paper we describe automorphisms of G ad (R, Φ) and E ad (R, Φ) (for the root systems A l , D l and E l the automorphisms were described in the paper [1]). Suppose that R is a local ring with 1/2 and 1/3. Then every automorphism of G ad (R, Φ) (E ad (R, Φ)) is standard (see below definitions of standard automorphisms). These results for Chevalley groups over fields were proved by R. Steinberg [2] for finite case and by J. E. Humphreys [3] for infinite case. K. Suzuki [4] studied automorphisms of Chevalley groups over rings of p-adic numbers. E. Abe [5] proved this result for Noetherian rings, but the class of all local rings is not completely contained in the class of Noetherian rings, and the proof of [5] can not be extended to the case of arbitrary local rings. From the other side, automorphisms of classical groups over rings were discussed by many authors. This field of research was open by Schreier and Van der Varden, who described automorphisms of the group P SL n (n ≥ 3) over arbitrary field. Then J. Diedonne [6], L. Hua and I. Reiner [7], O'Meara [8], B. R. McDonald [9], I. Z. Golubchik and A. V. Mikhalev [10] and others studied this problem for groups over more general rings. To prove our theorem we generalize some methods from the paper of V. M. Petechuk [11].Every Chevalley group under consideration is embedded into the group GL N (R) for some N ∈ N. Therefore we can consider Chevalley groups as matrix groups and use linear algebraic group technique: invertible coordinate changes in local rings, uniqueness of a solution of systems of linear equations over local rings with the condition, that determinant of a corresponding matrix is invertible, and so on. As the result we come to the fact that every automorphism of Chevalley group is induced by automorphism of the corresponding matrix ring.The author would like to thank A.Yu. Golubkov, A.A. Klyachko and A.V. Mikhalev for valuable comments and attention to the work.

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.…”

“…(5) if α + β ∈ Φ, and α, β are roots of the same length, then [x α , x β ] = cx α+β ; (6) if α + β ∈ Φ, α is a long root, β is a short root, then [x α , x β ] = ax α+β + bx α+2β .…”

Элементарная Эквивалентность Групп Шевалле Над Локальными Кольцами

“…Основные сведения о системах корней, полупростых алгебрах Ли, группах Шевалле, которые будут использоваться в настоящей работе, можно найти в работах автора [9]- [11]. Подробные сведения о системах корней и их свой-ствах можно найти в книгах [12], [13].…”

“…. , ξ l )r n ) ∧ (s Из работ [25], [10], [11] следует, что если две такие группы изоморфны, то кольца R и R ′ изоморфны. Если группы E и E ′ элементарно эквивалентны, то по теореме Кейслера-Шелаха (см.…”