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The purpose of this series of works is to show the existence of the strong Bernstein-Gelfand-Gelfand resolution (cf. [1]) of the irreducible highest weight module L(A) with dominant integral highest weight A over a symmetrizable generalized Kac-Moody algebra ( = GKM algebra). Here, GKM algebras are a class of contragredient Lie algebras Q(A) over C (see [4], [8], or [7, Chapter 1]) associated to a real square matrix A = (aij)ijel indexed by a finite set / which satisfies the conditions:

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