Enveloping algebras of infinite-dimensional Lie algebras

数学家对对称性很感兴趣，并且经常通过一种称为环的代数结构来建模对称性。“自然界中”遇到的大多数环都是非对易的：操作的顺序很重要。在真实的世界中，操作的顺序也很重要：先穿袜子再穿鞋子和先穿鞋子再穿袜子会产生不同的结果！不那么麻烦的是，在三维空间中移动时，顺序也很重要，这就是为什么大多数图形软件（如视频游戏和医学成像软件）使用称为四元数的非交换环进行计算。几何对象的对称性通常通过称为李代数的对象建模。反过来，李代数与称为包络代数的非交换环相关联。李代数通常通过它们的表示来研究，这些表示反映了李代数中编码的对称性。李代数和包络代数的性质往往微妙而有力地依赖于李代数表示的结构。数学家研究的通常几何对象有许多维度：例如，我们在其中移动的空间是三维的。然而，为了进行量子力学中涉及的精细而复杂的计算，物理学家需要研究具有无限多维的空间。Virasoro代数是一个著名的无限维李代数，它在数学和物理学中享有盛誉。它可以被看作是统计力学的数学模型，因此对物理学具有深刻的重要性。无限维李代数及其包络代数是出了名的难以理解。例如，近100年来，人们已经知道有限维李代数的包络代数具有一个称为“诺特”的性质，以德国数学家Emmy Noether命名。诺特环相对来说表现良好，而非诺特环则比较奇特。然而，没有人知道无限维李代数的包络代数是否可能是诺特代数。这个问题在45年前首次被提出，直到我在2013年证明了Virasoro李代数的包络代数不是noether，才取得了很小的进展。这个证明使用了Virasoro代数的几何表示，因此证明了几何技术在理解代数问题上的力量。这个提议的主要目的是证明无限维李代数不可能有诺特包络代数。我将通过各种方法来实现这一点，其中许多方法侧重于理解无限维李代数表示族的几何。理解这一点将应用于物理学以及数学的其他领域。

Mathematicians are interested in symmetry, and often model symmetry through an algebraic structure called a ring. Most rings encountered "in nature" are noncommutative: the order of operations matters. In the real world the order of operations also matters: putting on your socks before putting on your shoes gives a different result than putting on your shoes before your socks! Less frivolously, the order also matters when moving in three-dimensional space, which is why most graphics software (such as video games, and also medical imaging software) uses a noncommutative ring called the quaternions to do calculations.The symmetries of a geometric object are often modelled through an object called a Lie algebra. Lie algebras, in turn, are associated with noncommutative rings called enveloping algebras. Lie algebras are often studied through their representations, which echo the symmetry encoded in the Lie algebra. The properties of the Lie algebra and the enveloping algebra tend to depend, subtly and powerfully, on the structure of representations of the Lie algebra.The usual geometric objects that mathematicians study have finitely many dimensions: for example, the space we move around in is three-dimensional. In order to do the delicate and complicated calculations involved in quantum mechanics, however, physicists need to study spaces that have infinitely many dimensions. Their symmetries are encoded in infinite-dimensional Lie algebras.A famous infinite-dimensional Lie algebra is called the Virasoro algebra, which is renowned in mathematics and physics. It may be viewed as a mathematical model of statistical mechanics, and so is of deep importance to physics. Infinite-dimensional Lie algebras and their enveloping algebras are famously difficult to understand. For example, it has been known for almost 100 years that the enveloping algebras of finite-dimensional Lie algebras have a property called 'noetherian', named for the German mathematician Emmy Noether. Rings that are noetherian are relatively well-behaved; those that are not noetherian are more exotic. However, nobody knows if it is even possible for the enveloping algebra of an infinite-dimensional Lie algebra to be noetherian. This question was first asked in print 45 years ago, and very little progress had been made on it until I proved, in 2013, that the enveloping algebra of the Virasoro Lie algebra is not noetherian. This proof used the geometry of representations of the Virasoro algebra and so demonstrated the power of geometric techniques to understand algebraic problems.The main objective of this proposal is to prove that it is not possible for an infinite-dimensional Lie algebra to have a noetherian enveloping algebra. I will do this through a variety of methods, many focused on understanding the geometry of families of representations of infinite-dimensional Lie algebras. Understanding this will have applications to physics as well as other areas of mathematics.