  We present an inductive algebraic approach to the systematic construction and
classification of generalized Calabi-Yau (CY) manifolds in different numbers of
complex dimensions, based on Batyrev's formulation of CY manifolds as toric
varieties in weighted complex projective spaces associated with reflexive
polyhedra. We show how the allowed weight vectors in lower dimensions may be
extended to higher dimensions, emphasizing the roles of projection and
intersection in their dual description, and the natural appearance of
Cartan-Lie algebra structures. The 50 allowed extended four-dimensional vectors
may be combined in pairs (triples) to form 22 (4) chains containing 90 (91) K3
spaces, of which 94 are distinct, and one further K3 space is found using
duality. In the case of CY_3 spaces, pairs (triples) of the 10~270 allowed
extended vectors yield 4242 (259) chains with K3 (elliptic) fibers containing
730 additional K3 polyhedra. A more complete study of CY_3 spaces is left for
later work.
