The structure of non-Fricke Monstrous Lie algebras

TL;DR

This paper provides a structural analysis of a class of Borcherds algebras, including Monstrous Lie algebras related to non-Fricke elements of the Monster group, facilitating their computation.

Contribution

It introduces a structure theorem for certain Borcherds algebras and links them to Monstrous Lie algebras, offering a new approach to compute their twisted denominator formulas.

Findings

Decomposition of Borcherds algebras into free, Heisenberg, and abelian subalgebras

Identification of Monstrous Lie algebras associated with non-Fricke elements

Efficient method for computing twisted denominator formulas

Abstract

We consider Borcherds algebras with no real roots and the property that all zeroes in the Borcherds Cartan matrix occur in a single diagonal zero block. It follows that all other entries of the matrix are negative. We give a structure theorem for these Borcherds algebras, decomposing them into free, Heisenberg and abelian subalgebras. We show that a class of such Borcherds algebras are the Monstrous Lie algebras associated to non-Fricke elements of the Monster finite simple group. This new perspective on their structure gives an efficient method to compute their twisted denominator formulas.