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

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 En symmetry algebra can be extended either to a Borcherds superalgebra or to the Kac-Moody algebra E11, 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.