Manchester Geometry Seminar 2000/2001

We consider the problem of constructing a theory of Lorentzian (or hyperbolic) Kac--Moody algebras. This theory should be similar to the well-known theories of finite and affine Kac--Moody algebras.

As an example, we consider classificaiton of some interesting Lorentzian Kac--Moody algebras of rank three. Perhaps, this is the first example when a large class of Lorentzian Kac--Moody algebras was classified. This classification involves clasificaiton of hyperbolic root systems which are appropriate for the theory of Lorentzian Kac--Moody algebras, and classificaion of meromorphic reflective automorphic forms with infinite product with respect to the paramodular group (the modular group for Abelian surfaces).

See details in math.AG/9810001, 0010329