Enveloping algebras of infinite-dimensional Lie algebras

有限维李代数的包络代数可以说是研究得最充分的一般非交换环；事实上，环理论的许多技术最初都是在这种背景下发展起来的。相比之下，无限维李代数的包络代数，直到最近，几乎完全是神秘的，尽管这些李代数在数学物理和其他领域有着巨大的重要性。最近，Sierra博士和她的合作者在理解这些包络代数方面取得了多年来的第一个进展，证明了仿射线上衍生的Witt代数的包络代数不是左或右诺瑟的，并继续理解这种包络代数的增长及其因素。这个项目建立在Sierra博士的研究结果的基础上，以研究无限维李代数的其他例子，例如，研究一般代数变体的衍生李代数何时是诺etherian的。这个项目的结果将有助于解决一个普遍的问题，这个问题持续了五十年，即无限维李代数是否可能有一个诺etherian包络代数。

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.