Faces of polytopes and Koszul algebras

Abstract

Let $\g$ be a reductive Lie algebra and $V$ a $\g$-semisimple module. In this article, we study the category $\G$ of graded finite-dimensional representations of $\g \ltimes V$. We produce a large class of truncated subcategories, which are directed and highest weight. Suppose $V$ is finite-dimensional with weights $\wt(V)$. Let $Ψ\subset \wt(V)$ be the set of weights contained in a face $\F$ of the polytope that is the convex hull of $\wt(V)$. For each such $\Psi$, we produce quasi-hereditary Koszul algebras. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra $\spg$ of the locally finite part of the algebra of invariants $(END{\C} (\V) \otimes \Sym V)^{\g}$, where $\V$ is the direct sum of all simple finite-dimensional $\g$-modules. We prove that $\spg$ is Koszul of finite global dimension.

v3: Significant revisions. To appear in the Journal of Pure and Applied Algebra; 20 pages

References in corpus (1)

• Current algebras, highest weight categories and quivers

Cited by in corpus (7)

• Faces and maximizer subsets of highest weight modules
• Koszul duality for semidirect products and generalized Takiff algebras
• Standard parabolic subsets of highest weight modules
• String diagrams and categorification
• On multigraded generalizations of Kirillov-Reshetikhin modules
• Moving between weights of weight modules
• Tilting Modules in Truncated Categories