Symmetries of Borcherds algebras

TL;DR

This paper reviews the construction and properties of Borcherds algebras, especially the Monstrous Lie algebras linked to the Monster group, and explores their applications in string theory and related open problems.

Contribution

It provides an overview of Borcherds algebras, details the construction of Monstrous Lie algebras, and discusses recent developments in associating Lie groups to these algebras.

Findings

Monstrous Lie algebras relate to the Monster group.

Connections to Heterotic String models are discussed.

Open problems in the theory are summarized.

Abstract

We give an overview of the construction of Borcherds algebras, particularly the Monstrous Lie algebras $\mathfrak m_g$ constructed by Carnahan, where $g$ is an element of the Monster finite simple group. When $g$ is the identity element, $\mathfrak m_g$ is the Monster Lie algebra of Borcherds. We discuss the appearance of the $\mathfrak m_g$ in compactified models of the Heterotic String. We also summarize recent work on associating Lie group analogs to the Lie algebras $\mathfrak m_g$. We include a discussion of some open problems.