Weakly Noetherian Lie Algebra and the Sierra-Walton Conjecture

TL;DR

This paper introduces the concept of weakly Noetherian Lie algebras, classifies certain graded cases, and advances the Sierra-Walton conjecture by proving it for perfect graded Lie algebras.

Contribution

It defines weakly Noetherian Lie algebras, provides a classification for graded cases, and proves the Sierra-Walton conjecture for perfect graded Lie algebras.

Findings

Weakly Noetherian Lie algebras have a constrained structure.

Explicit classification of perfect strictly weakly Noetherian graded Lie algebras.

Proves the Sierra-Walton conjecture for all perfect graded Lie algebras.

Abstract

Let K be a field of characteristic zero. Motivated by the conjecture that an enveloping algebra U(g) is Noetherian only if g is finite dimensional, we define the notion of weakly Noetherian Lie algebras. The main result, Theorem A, states that weakly Noetherian Lie algebras have a very constrained structure. In the specific case of graded Lie algebras, it implies an explicit classification of the perfect strictly weakly Noetherian Lie algebras, stated in Theorem B. The proofs of both theorems are quite long, and uses concrete results due to Tits, Formanek, Razmyslov, Grabowski and the author. The first theorem provides some insight on the desired conjecture. The second one implies the conjecture for all perfect graded Lie algebras, improving a celebrated theorem of Sierra and Walton.