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

Abstract

In the given paper we prove that every automorphism of a Chevalley group of type B l , l 2, over a commutative local ring with 1/2 is standard, i. e., it is a composition of ring, inner and central automorphisms.

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

“…A similar result for local rings without 1/2 was obtained in [5]. In [6], similar statements for the root system F 4 were proved, and in [7] all previous results with the help of localization method were generalized for the case of adjoint Chevalley groups over arbitrary commutative rings.…”

Automorphisms of Chevalley groups of different types over commutative rings

“…It is precisely the situation of papers [17], [19], [16], [18], [20], where for a local ring S and root systems A 2 , B l , C l , F 4 for 2 ∈ S * , G 2 for 2, 3 ∈ S * , the root systems A l , l 3, D l , E 6 , E 7 , E 8 without any additional conditions it was proved that if in the group E ad (Φ, S) some elements x ′ α are the images of the corresponding x α (1), α ∈ Φ, and also…”

“…Let us now deal with the root system B 2 . Recall that in this system there are roots ±α 1 = ±(e 1 − e 2 ), ±α 2 = ±e 2 , ±α 3 = ±(α 1 + α 2 ) = ±e 1 , ±α 4 = ±(α 1 + 2α 2 ) = ±(e 1 + e 2 ) (detailed matrices for this system can be found in the papers [16] and [18]).…”

“…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.…”

The R∞-property for Chevalley groups of types Bl, Nasybullov1 2016Journal of Algebra10080View full textAdd to dashboardCite

“…Using this fact, the proof of the case 3 literally repeats the proof of the case 2 after changing the matrix Z(T ) by the matrix 1 ⊕ Z(T ), and using the result [16] (instead of [14,15]) about the automorphism group of the Chevalley groups of the type B l . Theorem is proved.…”