Automophisms of Nil-Triangular Subrings of Algebras Chevalley Type $G_2$ Over Integral Domain. I

Abstract

Let $N\Phi(K)$ be the nil-triangular subalgebra of the Chevalley algebra over an associative commutative ring $K$ with the identity associated with a root system $\Phi$. This paper studies the well-known problem of describing automorphisms of Lie algebras and rings $N\Phi(K)$. Automorphisms of the Lie algebra $N\Phi(K)$ under restrictions $K=2K=3K$ on ring $K$ are described by Y. Cao, D. Jiang, J. Wang (2007). When passing from algebras to Lie rings, the group of automorphisms expands. Thus, the subgroup of central automorphisms is extended, i.e. acting modulo the center, ring automorphisms induced by automorphisms of the main ring are added. For the type $A_{n}$, a description of automorphisms of Lie rings $N\Phi(K)$ over $K$ was obtained by V.M. Levchuk (1983). Automorphisms of the Lie ring $N\Phi(K)$ are described by V.M. Levchuk (1990) for type $D_4$ over $K$, and for other types by A.V. Litavrin (2017), excluding types $G_2$ and $F_4$. In this paper we describe automorphisms of a nil-triangular Lie ring of type $G_2$ over an integrity domain $K$ under the following restrictions on $K$: either $K=2K=3K$, or $3K = 0$. To study automorphisms, the upper and lower central series described in this paper, are essentially used.

Keywords

• chevalley algebra
• nil-triangular subalgebra
• ring
• automorphism
• hypercentral automorphism