Borcherds and Kac-Moody extensions of simple finite-dimensional Lie algebras

TL;DR

This paper explores the relationship between Borcherds superalgebras and Kac-Moody algebras derived from simple finite-dimensional Lie algebras, showing they produce identical p-form spectra in a generalized setting.

Contribution

It demonstrates that Borcherds and Kac-Moody extensions of simple Lie algebras yield the same p-form spectrum, extending previous results from supergravity contexts.

Findings

Borcherds and Kac-Moody extensions produce identical p-form spectra.

Generalizes the relationship beyond maximal supergravity.

Provides a unified framework for algebraic extensions and spectrum derivation.

Abstract

We study the Borcherds superalgebra obtained by adding an odd (fermionic) null root to the set of simple roots of a simple finite-dimensional Lie algebra. We compare it to the Kac-Moody algebra obtained by replacing the odd null root by an ordinary simple root, and then adding more simple roots, such that each node that we add to the Dynkin diagram is connected to the previous one with a single line. This generalizes the situation in maximal supergravity, where the E(n) symmetry algebra can be extended to either a Borcherds superalgebra or to the Kac-Moody algebra E(11), and both extensions can be used to derive the spectrum of p-form potentials in the theory. We show that also in the general case, the Borcherds and Kac-Moody extensions lead to the same p-form spectrum of representations of the simple finite-dimensional Lie algebra.