Lie structures and chain ideal lattices

TL;DR

This paper explores Lie algebras with chain lattice ideals, emphasizing their role as fundamental components in larger Lie algebra structures, and introduces computational algorithms to construct and analyze such algebras.

Contribution

It highlights the significance of chain lattice Lie algebras in decompositions and provides algorithms for their construction and classification.

Findings

Large class of chain lattice Lie algebras identified

Algorithms produce natural gradings and parametric families

Supports various Lie algebra structures

Abstract

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to show that the number of Lie algebras of this class is large and they support other types of Lie structures. Beginning with general examples and algebraic decompositions, we focus on computational algorithms to build Lie algebras in which the lattice of ideals is a chain. The chain condition forces gradings on the nilradicals of this class of algebras. Our algorithms yield to several positive naturally graded parametric families of Lie algebras. Further generalizations and other kind of structures will also be discussed.