Automorphisms of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

Abstract

In this paper, we prove that every automorphism of a Chevalley group of type G2 over a commutative local ring without 1/2 is the composition of a ring automorphism and a conjugation by some matrix.

Automorphisms of Chevalley groups of different types over commutative rings

“…It is precisely the situation of the paper [10], in which for a local ring R and the root system G 2 if 3 ∈ R * without any additional conditions for the ring it is proved, that if in the group E ad (G 2 , R) some elements x ′ α are the images of the corresponding x α (1), α ∈ G 2 , and also…”

Normalizers of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

“…The given paper is a continuation of the paper [8], finalizing the result of the mentioned work. The aim of this paper is to prove that every automorphism of Chevalley groups of type G 2 over a commutative local ring without 1/2 is a composition of ring and inner automorphisms.…”

“…, c 6 for each of the four equations under consideration). This system satisfies the conditions of the linearization method for local rings described in [8]. Solving it by this method, we have Z = 0, a i,j = b i,j = c i,j = 0 for all possible index combinations, i.e., C = E, as required.…”

“…We completely follow all definitions and notions from [8]. Recall that R is some commutative local ring with 1.…”

“…Recall also that Φ is the root system G 2 , and G(R) = G(Φ, R) is the Chevalley group of type G 2 over R. The given group is adjoint (since the root system G 2 has only one weight lattice); also, over a local ring, it coincides with its elementary subgroup generated by all root unipotents The adjoint representation of the group G(R) is 14-dimensional, and all matrices representing the elements x α (t), w α (t), and h α (t) for simple roots can be found in Appendix to the paper [8].…”