University of Birmingham
Mathematics

One of the most fundamental concepts in all areas of algebra is that of simplicity. The core idea is a familiar one to anyone who has studied maths at any level, and it is essentially that an object is simple when there are no non-trivial smaller objects inside it. Indeed, the very reason we study prime numbers is that they are ‘simple’, i.e., they cannot be factorised into smaller numbers. This is emblematic of the way in which mathematics often works, where we study simpler objects in order to understand more complicated ones. This is also the motivation behind representation theory, where we try to understand more sophisticated algebraic objects by thinking of them from the viewpoint of linear algebra. Combining these approaches, determining simple representations becomes a key part of representation theory.

This research proposal concerns the case when the algebraic objects we are studying are Lie algebras over fields of prime characteristic. This has been an area long studied by many world-renowned mathematicians, and there has been renewed interest in this area of late, primarily focusing on the simple modules appearing from the representation theory of algebraic groups. My research would focus on the more general question of representations for Lie algebras, where there remain vast depths left to be explored.

Central to the representation theory of Lie algebras in positive characteristic are certain objects called ‘baby Verma modules.’ If we can determine how baby Verma modules are built out of simple modules we can reverse the process to obtain major new results about the simple modules, a significant leap forward in the representation theory of Lie algebras.

The key innovation in my research will be the development of certain categories and modules where this data can be encoded as generalisations of the so-called ‘p-canonical basis’. This approach has been wildly successful in the theory of algebraic groups, including in papers by Riche, Williamson, and Abe, and is ripe for further exploration in the context of Lie algebras.