Three Minimax Ideal Relations of Lie Algebras

TL;DR

This paper introduces new ideal concepts in finite-dimensional Lie algebras, establishing their existence, properties, and relationships with solvability and nilpotency, thus advancing the structural understanding of Lie algebra theory.

Contribution

It defines near perfect, upper bounded, and perfect radicals, proving their existence and characterizing their roles in Lie algebra structure theory.

Findings

Largest perfect ideal equals smallest ideal of derived series

Largest near perfect ideal equals smallest ideal of lower central series

Factor algebra by these radicals is solvable or nilpotent

Abstract

In this paper, we introduce near perfect ideals and upper bounded ideals, and study them as well as perfect ideals for finite dimensional Lie algebras. We show that the largest perfect ideal and the largest near perfect ideal of a finite dimensional Lie algebra always exist, and are equal to the smallest ideal of the derived series, and the smallest ideal of the lower central series, respectively. We call them the perfect radical and the near perfect radical of that Lie algebra, respectively. A nonzero Lie algebra is solvable if and only if its perfect radical is zero. The factor algebra of a Lie algebra by its perfect radical is solvable. A nonzero Lie algebra is nilpotent if and only if its near perfect radical is zero. The factor algebra of a Lie algebra by its near perfect radical is nilpotent. We also show that the smallest upper bounded ideal always exists, and is equal to the largest ideal of the upper central series. For a nilpotent Lie algebra, there is only one upper bounded ideal, i.e., the nilpotent Lie algebra itself.