The strong Bernstein-Gelfand-Gelfand resolution for generalized Kac-Moody algebras. I: The existence of the resolution (Q1318900)

This is the first part of two papers concerning the strong Bernstein-Gelfand-Gelfand (BGG) resolution of the irreducible highest weight module over a symmetrizable generalized Kac-Moody (GKM) algebra. This paper is devoted to proving the existence of the weak BGG resolution. The construction is carried out basically in the same way as in [\textit{H. Garland} and \textit{J. Lepowsky}, Invent. Math. 34, 37--76 (1976; Zbl 0358.17015)]. As an application a vanishing theorem for the Lie algebra homology is obtained, which is the GKM version of a result of \textit{S. Kumar} [J. Algebra 102, 444--462 (1986; Zbl 0633.17012)]. A GKM generalization of Bott's ``strange equality'' [\textit{J. Lepowsky}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 12, 169--234 (1979; Zbl 0414.17007)] is also proved.