Polytopal linear retractions

We investigate graded retracts of polytopal algebras (essentially the homog eneous rings of affine cones over projective toric varieties) as polytopal analogues of vector spaces. In many cases we show that these retracts are a gain polytopal algebras and that codimension 1 retractions factor through r etractions preserving the semigroup structure. We expect that these results hold in general. This paper is a part of the project started by the authors in 1999, where w e investigate the graded automorphism groups of polytopal algebras. Part of the motivation comes from the observation that there is a reasonable 'poly topal' generalization of linear algebra (and, subsequently, that of algebra ic K-theory).