Some Generalized Kac-Moody Algebras With Known Root Multiplicities

TL;DR

This paper constructs a sequence of generalized Kac-Moody algebras derived from Borcherds' fake monster Lie algebra, providing explicit descriptions of their root systems and multiplicities, and deriving bounds for hyperbolic Lie algebras.

Contribution

It introduces a new sequence of algebras with explicitly described root data, extending understanding of root multiplicities and structure in generalized Kac-Moody algebras.

Findings

Explicit formulas for denominator functions and root multiplicities.

Decomposition of root systems into convex holes of finite and affine type.

Strong upper bounds for root multiplicities of hyperbolic Lie algebras.

Abstract

Starting from Borcherds' fake monster Lie algebra we construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including AE3.