The universal enveloping algebra of the Witt algebra is not noetherian

TL;DR

This paper proves that the universal enveloping algebra of the Witt algebra, an infinite dimensional Lie algebra, is not noetherian, answering a long-standing open question in algebra.

Contribution

It establishes that the universal enveloping algebra of the Witt algebra and related infinite dimensional Lie algebras are not noetherian, using algebro-geometric techniques.

Findings

Universal enveloping algebra of the Witt algebra is not noetherian.

Enveloping algebras of many other infinite dimensional Lie algebras are also not noetherian.

Main result answers a 23-year-old open question in the field.

Abstract

This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces. As a consequence of our main result, we also show that the enveloping algebras of many other infinite dimensional Lie algebras are not noetherian. These Lie algebras include the Virasoro algebra and all infinite dimensional Z-graded simple Lie algebras of polynomial growth.