Polytopal linear retractions (Q2750949)

The category of finite-dimensional vector spaces over a field \(k\) is a full subcategory of that so-called polytopal \(k\)-linear category \({\mathcal P}ol_k\). The objects of this category being standard graded \(k\)-algebras associated with a lattice polytope and the morphisms the homogeneous \(k\)-algebra homomorphisms. NEWLINENEWLINENEWLINEThis paper is part of a project of the authors where they investigate the graded automorphism groups of polytopal algebras and, on them, they study retractions. In this sense, the authors prove several results which strongly support the following two conjectures: NEWLINENEWLINENEWLINE(1) The images of retractions in \({\mathcal P}ol_k\) are polytopal algebras. NEWLINENEWLINENEWLINE(2) A codimension 1 retraction factors through either a facet projection or an affine lattice retraction of the underlying lattice polytope.