On the fundamental representation of Borcherds algebras with one imaginary simple root

TL;DR

This paper investigates the fundamental representation of Borcherds algebras with one imaginary simple root, showing it is given by the full tensor algebra over an integrable highest weight module, with potential physical applications.

Contribution

It provides a detailed description of the fundamental representation for Borcherds algebras with a single imaginary simple root, extending understanding of their structure.

Findings

Irreducible module is the full tensor algebra over an integrable highest weight module.

Analysis of Borcherds algebras with one imaginary simple root.

Discussion on physical realizations as symmetry algebras in quantum field theory.

Abstract

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, and the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary simple roots. The simplest nontrivial examples one can think of are those where one adds ``by hand'' one imaginary simple root to an ordinary Kac-Moody algebra. We study the fundamental representation of this class of examples and prove that an irreducible module is given by the full tensor algebra over some integrable highest weight module of the underlying Kac-Moody algebra. We also comment on possible realizations of these Lie algebras in physics as symmetry algebras in quantum field theory.