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Faces of Weight Polytopes, a Generalization of a Theorem of Vinberg and Koszul Algebras


Let $\g$ be a reductive Lie algebra over $\C$ and let $V$ be a $\g$-semisimple module. In this article, we study the category $\ghat$ of $\Z_+$-graded $\g\ltimes V$-modules with finite-dimensional grade pieces. We construct and classify certain special subsets called {\it 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 {\it 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 $\ghat$ which are directed and highest weight.

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