Root systems and Lie Algebras, Lattices and Vertex Algebras
Evan Williams Theatre
The classification of simple Lie algebras by root systems is a jewel of representation theory, and the classification of integral lattices is an important problem with links to modular forms and practical applications.
I will argue that a natural common generalisation of these two problems is the classification of rational vertex algebras; algebras which axiomatise part of the structure of quantum field theory and which play an important role in modern representation theory.
I will mostly focus on the self-dual rational vertex algebras of rank 24, these form an especially beautiful class which includes the algebra responsible for monstrous moonshine. Based on the pioneering work of the physicist A. N. Schellekens, these algebras are conjectured to be exactly 71 in number, and their classification depends on an interesting blend of techniques from Lie algebras, integral lattices, and modular forms.
Parts of this talk are based on joint work with C.-H. Lam, S. Moeller, N. Scheithauer, and H. Shimakura.
Dr Jethro van Ekeren, Universidade Federal Fluminense, Niteroi