Jacobson program for Lie algebras

TL;DR

This paper explores the structure of Lie algebras by introducing Lie-modules, characterizing primitive Lie algebras, and emphasizing the role of nuclear ideals in understanding their properties.

Contribution

It develops a structure theory for Lie algebras using Lie-modules and introduces new characterizations and representation theorems for primitive Lie algebras.

Findings

Characterization of primitive Lie algebras via Lie-modules

Representation theorem for Lie algebras using primitive Lie algebras

Importance of nuclear ideals in prime and semiprime Lie algebras

Abstract

Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a Lie algebra. We give several Lie-module-related characterizations of primitive Lie algebras and prove a representation theorem of a Lie algebra in terms of primitive Lie algebras. We prove a density theorem. Finally, we highlight on the importance of nuclear ideals of prime and semiprime Lie algebras in determining them.