The faithful enveloping rings of the nil-triangular ring of type $G_2$ and their automorphisms

A. V. Kazakova

Siberian Federal University (Krasnoyarsk)

Abstract: The structure of the Chevalley algebra over a field or ring $K$, associated with an indecomposable root system $\Phi$, essentially depends on its nil-triangular subalgebra $N\Phi(K)$. It turned out to be natural for $N\Phi(K)$ to use the faithful enveloping algebra $R$, introduced in 2018, which has the same basis as $N\Phi(K)$. It is known that the isomorphism of the Lie rings $N\Phi(K)$ does not depend on the choice of signs of the structure constants $N_{r, s}$. However, for faithful enveloping rings $R$ this property is violated. Therefore, the question of describing their automorphisms was extended to finding all non-isomorphic faithful enveloping rings $N\Phi(K)$ of type $G_2$ over $K$, and only then finding an explicit description of their automorphisms.

Keywords: Lie algebra, Chevalley algebra, faithful enveloping algebra, nil-triangular subalgebra, standard automorphism, upper central series, hypercentral automorphism.

UDC: 512.554

Received: 14.03.2024
Accepted: 04.09.2024

DOI: 10.22405/2226-8383-2024-25-3-118-142