Related papers: From forced gradings to Q-Koszul algebras

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…

Let $G$ be a simple, simply connected algebraic group over an algebraically closed field of positive characteristic $p$. In recent work, the authors have studied a graded analogue of the category of rational $G$-modules. These gradings are…

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…

In previous work, the authors introduced the notion of Q-Koszul algebras, as a tool to "model" module categories for semisimple algebraic groups over fields of large characteristics. Here we suggest the model extends to small…

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…

Given a quiver with potential $(Q,W)$, Kontsevich-Soibelman constructed a Hall algebra on the cohomology of the stack of representations of $(Q,W)$. As shown by Davison-Meinhardt, this algebra comes with a filtration whose associated graded…

Let $G$ be a semisimple, simply connected algebraic group defined and split over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a projective…

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the…

The main purpose of this paper is computing higher algebraic $K$-theory of Koszul complexes over principal ideal domains. The second purpose of this paper is giving examples of comparison techniques on algebraic $K$-theory for Waldhausen…

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…

For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…

This paper examines (restricted) Koszul Lie algebras, a class of positively graded Lie algebras with a quadratic presentation and specific cohomological properties. The study employs HNN-extensions as a key tool for decomposing and…

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…

Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others.…

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…

We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings…

We give a class of finite subgroups $G<SL(n, k)$ for which the skew-group algebra $k[x_1,\ldots, x_n]\#G$ does not admit the grading structure of a higher preprojective algebra. Namely, we prove that if a finite group $G<SL(n, k)$ is…

There are many structures (algebras, categories, etc) with natural gradings such that the degree 0 components are not semisimple. Particular examples include tensor algebras with non-semisimple degree 0 parts, extension algebras of standard…

We provide $\mathbb{N}$-filtrations on the negative part $U_q(\mathfrak{n}^-)$ of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra…

We introduce the notion of asymptotic universal Koszulity for graded-commutative algebras generated in degree~$1$, capturing the idea that an infinite-dimensional algebra can be approximated by a filtered system of finite-type universally…