Faces of polytopes and Koszul algebras

Let g be a semisimple Lie algebra and V a g-semisimple module. In this paper, we study the category G of Z-graded finite-dimensional representations of g⋉V. We show that the simple objects in this category are indexed by an interval-finite poset and produce a large class of truncated subcategories which are directed and highest weight. In the case when V is a finite-dimensional g-module, we construct a family of Koszul algebras which are indexed by certain subsets of the set of weights wt(V) of V. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra of the locally finite part of the algebra of invariants , where is the direct sum of all simple finite-dimensional g-modules. We prove that is Koszul of finite global dimension.