Related papers: Koszulity for semi-infinite highest weight categor…

Let A be a Koszul algebra, and $mod A$ the category of finitely generated graded left A-modules. The "linearity defect" ld_A(M) of $M \in mod A$ is an invariant defined by Herzog and Iyengar. An exterior algebra E is a Koszul algebra which…

The aim of this paper is to define and study some quasi-hereditary covers for higher zigzag algebras. We will show how these algebras satisfy three different Koszul properties: they are Koszul in the classical sense, standard Koszul and…

We classify blocks of categories of weight and generalized weight modules of algebras of twisted differential operators on P^n. Necessary and sufficient conditions for these blocks to be tame and proofs that some of the blocks are Koszul…

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…

The main result of this paper is to establish precisely which blocks in the Category $\mathcal{O}$ of the periplectic Lie superalgebra $\mathfrak{pe}(2)$ are Koszul. It is known that $\mathcal{O}(\mathfrak{pe}(2))$ has three blocks up to…

We give a description of the connected graded algebras which are finitely generated and presented of global dimension 2 or 3 and which are Gorenstein. These algebras are constructed from multilinear forms. We generalize the construction by…

We prove a highest weight theorem classifying irerducible finite--dimensional representations of quantum affine algebras and survey what is currently known about the structure of these representations.

This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. To this purpose, we endow the Koszul dual category of curved…

We extend the bar-cobar adjunction to operads and properads, not necessarily augmented. Due to the default of augmentation, the objects of the dual category are endowed with a curvature. We handle the lack of augmentation by extending the…

Given a commutative algebra $\mathcal O$, a proper ideal $\mathcal I$, and a resolution of $\mathcal O/ \mathcal I$ by projective $\mathcal O $-modules, we construct an explicit Koszul-Tate resolution. We call it the arborescent Koszul-Tate…

We study a properly convex real projective manifold with (possibly empty) compact, strictly convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact…

Given any Koszul algebra of finite global dimension one can define a new algebra, which we call a higher zigzag algebra, as a twisted trivial extension of the Koszul dual of our original algebra. If our original algebra is the path algebra…

Let $A$ be a finite dimensional $k$-algebra standardly stratified for a partial order $\leqslant$ and $\Delta$ be the direct sum of all standard modules. In this paper we study the extension algebra $E= \text{Ext}_A^{\ast} (\Delta, \Delta)$…

We define a basic class of algebras which we call homotopy path algebras. We find that such algebras always admit a cellular resolution and detail the intimate relationship between these algebras, stratifications of topological spaces, and…

We prove that the cone over a Dirichlet arrangement is supersolvable if and only if its Orlik-Solomon algebra is Koszul. This was previously shown for four other classes of arrangements. We exhibit an infinite family of cones over Dirichlet…

This is the first of four articles studying some slight generalisations H(n,m) of Khovanov's diagram algebra, as well as quasi-hereditary covers K(n,m) of these algebras in the sense of Rouquier, and certain infinite dimensional limiting…

Let V be the Veronese cubic surface in P^9. We classify the projections of V to P^8 whose coordinate rings are Koszul. In particular we obtain a purely theoretical proof of the Koszulness of the pinched Veronese, a result obtained…

The aim of this sequel to arXiv:1812.02935 is to set up the cornerstones of Koszul duality and Koszulity in the context of operads over a large class of operadic categories. In particular, for these operadic categories we will study…

We prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of $\Tor_n^A(R,R)$. As an application of these…

We define complete Segal objects, which play the role of internal higher category objects. Then we study them using representable Cartesian fibrations, in particular defining adjunctions and limits of complete Segal objects. Finally we use…