A Theory of Lorentzian Kac-Moody algebras

TL;DR

This paper develops a generalized theory of Lorentzian Kac-Moody algebras, connecting automorphic forms, hyperbolic lattices, and Weyl groups, extending classical and affine Kac-Moody algebra theories.

Contribution

It introduces a new framework for Lorentzian Kac-Moody algebras using Borcherds' generalized algebras and automorphic forms, advancing classification and structural understanding.

Findings

Presented a variant of Lorentzian Kac-Moody algebra theory

Connected denominator functions with automorphic forms on Hermitian domains

Achieved classification results and constructed examples of Lorentzian Kac-Moody algebras

Abstract

We present a variant of the Theory of Lorentzian (i. e. with a hyperbolic generalized Cartan matrix) Kac-Moody algebras recently developed by V. A. Gritsenko and the author. It is closely related with and strongly uses results of R. Borcherds. This theory should generalize well-known Theories of finite Kac-Moody algebras (i. e. classical semisimple Lie algebras corresponding to positive generalized Cartan matrices) and affine Kac-Moody algebras (corresponding to semi-positive generalized Cartan matrices). Main features of the Theory of Lorentzian Kac-Moody algebras are: One should consider generalized Kac-Moody algebras introduced by Borcherds. Denominator function should be an automorphic form on IV type Hermitian symmetric domain (first example of this type related with Leech lattice was found by Borcherds). The Kac-Moody algebra is graded by an integral hyperbolic lattice $S$. Weyl group acts in the hyperbolic space related with $S$ and has a fundamental polyhedron $\Cal M$ of finite (or almost finite) volume and a lattice Weyl vector. There are results and conjectures which permit (in principle) to get a ``finite'' list of all possible Lorentzian Kac-Moody algebras. Thus, this theory looks very similar to Theories of finite and affine Kac-Moody algebras but is much more complicated. There were obtained some classification results on Lorentzian Kac-Moody algebras and many of them were constructed.