Related papers: Faces of polytopes and Koszul algebras

Let $\lie g$ be a simple Lie algebra and let $\bs^{\lie g}$ be the locally finite part of the algebra of invariants $(_\bc\bv\otimes S(\lie g))^{\lie g}$ where $\bv$ is the direct sum of all simple finite-dimensional modules for $\lie g$…

Let $\mathfrak g$ be a complex simple Lie algebra and let $\Psi$ be an extremal set of positive roots. One associates with $\Psi$ an infinite dimensional Koszul algebra $\bold S_\Psi^{\lie g}$ which is a graded subalgebra of the locally…

The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma…

We find for each simple finitary Lie algebra $\mathfrak{g}$ a category $\mathbb{T}_\mathfrak{g}$ of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in…

In this paper we study representations of skew group algebras $\Lambda G$, where $\Lambda$ is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field $k$ with characteristic $p…

In this paper we study general highest weight modules $\mathbb{V}^\lambda$ over a complex finite-dimensional semisimple Lie algebra $\mathfrak{g}$. We present three formulas for the set of weights of a large family of modules…

This paper has two parts. The main goal, carried out in Part I, is to survey some recent work by the authors in which "forced" grading constructions have played a significant role in the representation theory of semisimple algebraic groups…

We study a semisimple extension of a Takiff superalgebra which turns out to have a remarkably rich representation theory. We determine the blocks in both the finite-dimensional and BGG module categories and also classify the Borel…

It is shown that, the quasi-Koszulities of algebras and modules are Morita invariance. A finite-dimensional $K$-algebra $A$ with an action of $G$ is quasi-Koszul if and only if so is the skew group algebra $A \ast G$, where $G$ is a finite…

Let $G$ be a semisimple algebraic group over an algebraically closed field $k$ of positive characteristic $p$. Under some restrictions on the size of $p$, the present paper establishes new results on the $G$-module structure of…

We discover a new connection between Koszul theory and representation theory. Let $\La$ be a quadratic algebra defined by a locally finite quiver with relations. Firstly, we give a combinatorial description of the local Koszul complexes and…

Motivated by the representation theory of symplectic reflection algebras, deformed preprojective algebras, and graded Hecke algebras, we consider filtered algebras $U$ whose associated graded is Koszul. The Koszul dual of $U$, as defined by…

Let $V$ be a highest weight module over a Kac-Moody algebra $\mathfrak{g}$, and let conv $V$ denote the convex hull of its weights. We determine the combinatorial isomorphism type of conv $V$, i.e. we completely classify the faces and their…

We prove that the category of graded finitely generated representations of the the cyclotomic quiver Schur algebra is a Koszul category.

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on…

Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good…

Chari and Greenstein [Adv. Math. 2009] introduced combinatorial subsets of the roots of a finite-dimensional simple Lie algebra $\mathfrak{g}$ which were important in studying Kirillov-Reshetikhin modules over $U_q(\widehat{\mathfrak{g}})$…

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody…

In this article we define $G$-algebras, that is, graded algebras on which a reductive group $G$ acts as gradation preserving automorphisms. Starting from a finite dimensional $G$-module $V$ and the polynomial ring $\mathbb{C}[V]$, it is…

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…