The finite minuscule and d-complete posets generalize Young and shifted Young diagrams.  They have many nice combinatorial properties; for example, minuscule posets are Gaussian and Sperner, and d-complete posets have the hook length and jeu de taquin properties.  Infinite analogs of the colored minuscule posets were used by R.M. Green to construct representations of many affine Kac-Moody algebras.  In this talk, we will present our recent characterizations of all such poset-built representations of Kac-Moody derived algebras and their “Borel derived” subalgebras, which led to new cardinality-independent definitions of colored minuscule and d-complete posets.  This is the first definition given for infinite colored d-complete posets.  We will also present the classifications of these posets.