Enveloping algebras of infinite-dimensional Lie algebras

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.

The primary impact of this project will be on developing and maintaining expertise in the UK in the fields of noncommutative ring theory, Lie theory, and representation theory within abstract algebra. These are areas where UK mathematics is strong, and this project will increase the internationally prominent role that UK mathematics plays in these fields. The project proposes to investigate problems of international relevance and interest. This will maintain and increase the world-leading role of UK researchers in these areas, which has been recognised in the past two international reviews of mathematics.

A key part of the proposed grant will be the training of two postdoctoral researchers (PDRAs). Thus this project will have a direct impact on building the capacity of the UK in the mathematical sciences and in STEM fields more generally.

Mathematics is fundamental to the UK economy as explained in a recent study by EPSRC. The study showed that "[t]he quantified contribution of mathematical science research to the UK economy in 2010 is estimated to be approximately 2.8 million in employment terms (around 10 per cent of all jobs in the UK) and £208 billion in terms of GVA contribution (around 16 per cent of total UK GVA)." As well as these direct impacts, mathematical research activities by organisations and employees also has indirect and induced effects. The 2018 EPSRC report "The Era of Mathematics" (the Bond Report) found that mathematical sciences research produced a return on investment of 588 to 1 in terms of economic benefit to the UK economy.

Research in pure mathematics is of benefit to society more broadly. As the 2010 International Review of Mathematical Sciences said, "the mathematical sciences provide a universal language for expressing abstractions in science, engineering, industry and medicine; mathematical ideas, even the most theoretical, can be useful or enlightening in unexpected ways, sometimes several decades after their appearance; the mathematical sciences play a central role in solving problems from every imaginable application domain; and, because of the unity of the mathematical sciences, advances in every sub-area enrich the entire field." Thus this research will have distributed impacts which are hard to quantify but are nonetheless profound.