Root systems for Lorentzian Kac-Moody algebras in rank 3 (arXiv version)

TL;DR

This paper classifies rank 3 hyperbolic root systems of Lorentzian Kac-Moody algebras that meet specific automorphic correction conditions, identifying 994 cases with up to 24 simple roots, revealing potential for discovering complex structures.

Contribution

It provides a comprehensive classification of hyperbolic rank 3 root systems satisfying automorphic correction conditions, expanding understanding of Lorentzian Kac-Moody algebra structures.

Findings

994 classified root systems with finite simple roots

Maximum of 24 simple roots in these systems

Patterns suggest some complex cases may be particularly rich

Abstract

Sometimes a hyperbolic Kac-Moody algebra admits an automorphic correction, meaning a generalized Kac-Moody algebra with the same real simple roots and whose denominator function has good automorphic properties; these for example allow one to work out the root multiplicities. Gritsenko and Nikulin have formalized this in their theory of Lorentzian Lie algebras and shown that the real simple roots must satisfy certain conditions in order for the algebra to admit an automorphic correction. We classify the hyperbolic root systems of rank 3 that satisfy their conditions and have only finite many simple roots, or equivalently a timelike Weyl vector. There are 994 of them, with as many as 24 simple roots. Patterns in the data suggest that some of the non-obvious cases may be the richest.