Generalized Kac-Moody algebras and the modular functionj

Abstract

The generalized Kac-Moody algebras were introdcued by Borcherds in his study of the vertex algebras and Monstrous Moonshine [B1-B3]. The structure and the representation theory of generalized Kac-Moody algebras are very similar to those of Kac-Moody algebras, and a lot of facts about Kac-Moody algebras can be generalized to generalized Kac-Moody algebras. The main difference is that the generalized Kac-Moody algebras may have imaginm 3, simple roots with norms __ 1. For unitarizable irreducible highest weight modules over symmetrizable generalized Kac-Moody algebras, Borcherds proved a character formula [B2], called the Weyl-Kac-Borcherds formula, which generalizes the WeylKac formula for integrable highest weight modules over symmetrizable Kac-Moody algebras [K]. As in the case of Kac-Moody algebras, the Weyl-Kac-Borcherds formula, when applied to the l-dimensional trivial representation, yields the denominator identio,. In this paper, using the Weyl-Kac-Borcherds formula and the denominator identity, we obtain a closed form root multiplicity formula for all symmetrizable generalized Kac-Moody algebras (Theorem 2.1). Our formula enables us to study the structure of a symmetrizable generalized Kac-Moody algebra 9 as a representation of a Kac-Moody algebra 0o contained in ~. The choice of ~0 gives rise to various expressions of the root multiplicities of 13, which would yield some interesting combinatorial identities. In Sect. 3, we apply this principle to a rank 2 generalized Kac-Moody algebra to obtain a combinatorial identity for the Pascal numbers. In [B3], Borcherds proved the Moonshine conjecture [CNI by constructing the Monster Lie algebra using the Monster vertex algebra [FLM] and the vertex algebra associated with the 2-dimensional even Lorentzian lattice IIl,~ [B 1]. The Monster Lie