Generalized Kac-Moody algebras and the modular function \(j\) (Q1318138)

We obtain a closed form root multiplicity formula for all symmetrizable generalized Kac-Moody algebras using the Weyl-Kac-Borcherds character formula and the denominator identity. Our formula enables us to study the structure of a symmetrizable generalized Kac-Moody algebra \({\mathfrak g}\) as a representation of a Kac-Moody algebra \({\mathfrak g}_ 0\) contained in \({\mathfrak g}\). Applying our formula to the Monster Lie algebra, we obtain some interesting relations for the coefficients of the elliptic modular function \(j\).