Enveloping algebras of infinite-dimensional Lie algebras

Enveloping algebras of finite-dimensional Lie algebras are, arguably, the most well-studied general class ofnoncommutative rings; indeed, many of the techniques of ring theory were first developed in this context. Incontrast, enveloping algebras of infinite-dimensional Lie algebras have, until recently, been almost entirelymysterious, in spite of the huge importance of these Lie algebras in mathematical physics and other areas.Recently, Dr Sierra and her collaborators have made the first progress for many years in understanding theseenveloping algebras, proving that the enveloping algebra of the Witt algebra of derivations on the affine line isnot left or right noetherian and going on to understand the growth of this enveloping algebra and its factors. Thisproject builds on Dr Sierra's results to investigate other examples of infinite-dimensional Lie algebras,investigating, for example, when Lie algebras of derivations on general algebraic varieties are noetherian. The results of this project will help to settle the general question, open for fifty years, of whether it is possible for aninfinite-dimensional Lie algebra to have a noetherian enveloping algebra.