Noetherian enveloping algebras of simple graded Lie algebras

TL;DR

This paper proves that the universal enveloping algebra of a simple graded Lie algebra is Noetherian only if the Lie algebra is finite-dimensional, establishing a key link between algebraic properties and dimensionality.

Contribution

It demonstrates a necessary condition for the Noetherian property of enveloping algebras of simple graded Lie algebras, connecting algebraic structure with dimensional constraints.

Findings

Universal enveloping algebra is Noetherian only for finite-dimensional simple graded Lie algebras.

Provides a criterion linking Noetherian property to the dimension of the Lie algebra.

Advances understanding of the structure of graded Lie algebras and their enveloping algebras.

Abstract

It is shown that if the universal enveloping algebra of a simple $\mathbb Z^n$-graded Lie algebra is Noetherian, then the Lie algebra is finite-dimensional.