Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

TL;DR

This paper introduces a new class of Lorentzian Kac-Moody algebras associated with 2-reflective hyperbolic lattices, classifies these lattices, and constructs automorphic forms that serve as their denominator identities.

Contribution

It classifies 2-reflective hyperbolic lattices with finite volume reflection groups and constructs automorphic forms as Borcherds products for associated Lorentzian Kac-Moody algebras.

Findings

Classification of 2-reflective hyperbolic lattices with finite volume groups

Construction of automorphic forms as Borcherds products

Explicit examples of automorphic corrections for Lorentzian Kac-Moody algebras

Abstract

We describe a new large class of Lorentzian Kac--Moody algebras. For all ranks, we classify 2-reflective hyperbolic lattices S with the group of 2-reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac--Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac--Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2-reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.