A resolution for standard modules of Borcherds Lie algebras

In this paper we construct a Bernstein–Gelfand–Gelfand-type resolution of a standard (irreducible) module for a Borcherds Lie algebra image. The resolution involves generalized Verma modules; the “reductive part” of the “parabolic subalgebra” of image may be a Borcherds algebra. This resolution, which generalizes that of Garland and Lepowsky for Kac–Moody algebras, is constructed here in sufficient generality to include the cases of the monster and fake monster Lie algebras. These algebras are of particular interest because of their connection to vertex algebras and conformal field theory. The monster Lie algebra is of importance because it is constructed from the moonshine module vertex algebra, appearing in the study of “monstrous moonshine” relating to the Fischer–Griess Monster simple group. The resolution constructed in this paper is used to obtain Kostantʹs homology formula for Borcherds algebras, and the character formula for a standard module.