Faces of weight polytopes, a generalization of a theorem of Vinberg and Koszul algebras

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