# The universal enveloping algebra of the Witt algebra is not noetherian

SPEAKER: Chelsea Walton

TITLE: The universal enveloping algebra of the Witt algebra is not noetherian.

ABSTRACT: This talk 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. This is achieved by establishing our main result: the universal

enveloping algebra of the positive part of the Witt algebra is not noetherian.

To this end, we employ algebro-geometric techniques from Sue Sierra’s

classification of (noncommutative) birationally commutative projective surfaces.

As a consequence of our main result, we show that the enveloping algebras of

many other infinite dimensional Lie algebras also are not noetherian. These

Lie algebras include the Virasoro algebra and all Z-graded simple Lie algebras

of polynomial growth.

This is joint work with Sue Sierra.