Vinberg’s Algorithm and Kac-Moody algebras

Abstract: Vinberg’s algorithm was introduced by Vinberg in order to calculate the fundamental domains of hyperbolic reflection groups, especially those coming from Lorentzian lattices. We will show how to use it to calculate the automorphism groups of some lattices, culminating in Conway’s spectacular discovery that the Dynkin diagram of the 26-dimensional even unimodular Lorentzian lattice is the Leech lattice. We will then discuss some of the Kac-Moody algebras associated with Vinberg’s Dynkin diagrams.

Mathematics

Audience: researchers in the discipline

The Vinberg Lecture Series