| Let g be a reductive Lie algebra over C and let V be a g -semisimple module. In this article, we study the category G&d14; of Z+ -graded g⋉ V-modules with finite-dimensional grade pieces. We construct and classify certain special subsets called weak F -faces of the set of weights of V. If V is a generalized Verma module, our result allows us to recover and extend a result due to Vinberg on the classification of faces of the weight polytope.;If g is semisimple and V is simple, we use the positive weak F -faces of the set of weights of V to construct a large family of Koszul algebras which have finite global dimension. We are also able to construct truncated subcategories of G&d14; which are directed and highest weight. |
