Peter Connick L&S Sciences
Investigating Root-Multiplicities of Kac-Moody Algebras
Over the summer, we propose to investigate the root multiplicities of (generalized) Kac-Moody Algebras. Our plan is to create an open-source computer package that allows for the computation of root multiplicities of Kac-Moody algebras, building upon the existing tools available to computational mathematicians, for instance, the popular library sage-math. Once we have developed and verified this package against known tables of root multiplicities, we aim to start investigating the root multiplicities of simple graphs, and attempt to address some outstanding conjectures on the distributions of root multiplicities. A greater understanding of the root multiplicities of Kac-Moody algebras would help with questions such as finding natural geometrical realizations of the algebras, and shed light on the connections that have already been found with mathematical physics. Some of these applications may include generalised knot invariants defined over Kac-Moody algebras rather than Lie Algebras, or the study of symmetries of systems and spaces that are not finite-dimensional.