Symmetries of Borcherds algebras

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.