Derived categories of graded gentle one-cycle algebras

Let A be a graded algebra. It is shown that the derived category of dg modules over A (viewed as a dg algebra with trivial differential) is a triangulated hull of a certain orbit category of the derived category of graded A-modules. This is applied to study derived categories of graded gentle one-cycle algebras.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.     

Skew differential operator algebras of twisted Hopf algebras

By introducing a twisted Hopf algebra we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-positive part of a quantized generalized Kac-Moody algebra, by identifying the canonical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group like.     

A homological characterization of piecewise hereditary algebras

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite. Dedicated to Otto Kerner on the occasion of his 65th birthday. The second author is supported by a grant from NSA. These results were obtained during a visit of the first author at Syracuse University. He would like to thank the second author for the hospitality during his stay.     

Bounds on the global dimension of certain piecewise hereditary categories

We give bounds on the global dimension of a finite length, piecewise hereditary category in terms of quantitative connectivity properties of its graph of indecomposables.We use this to show that the global dimension of a finite-dimensional, piecewise hereditary algebra A cannot exceed 3 if A is an incidence algebra of a finite poset or more generally, a sincere algebra. This bound is tight.     

The almost split triangles for perfect complexes over gentle algebras

Throughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N₊, we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i,j∈Z, [i,j]:={l∈Z∣i≤l≤j} (in particular, [i,j]=∅ if i>j).     

Combinatorial derived invariants for gentle algebras

We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable Auslander-Reiten quiver of its repetitive algebra. As a by-product we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.     

Tilting via torsion pairs and almost hereditary noetherian rings

We generalize the tilting process by Happel, Reiten and Smalø to the setting of finitely presented modules over right coherent rings. Moreover, we extend the characterization of quasi-tilted artin algebras as the almost hereditary ones to all right noetherian rings.     

Derived equivalences and Gorenstein algebras

We introduce a notion of Gorenstein R-algebras over a commutative Gorenstein ring R. Then we provide a necessary and sufficient condition for a tilting complex over a Gorenstein R-algebra A to have a Gorenstein R-algebra B as the endomorphism algebra and a construction of such a tilting complex. Furthermore, we provide an example of a tilting complex over a Gorenstein R-algebra A whose endomorphism algebra is not a Gorenstein R-algebra.     

Cohomological properties of the quantum shuffle product and application to the construction of quasi-Hopf algebras

For a commutative algebra the shuffle product is a morphism of complexes. We generalize this result to the quantum shuffle product, associated to a class of non-commutative algebras (for example all the Hopf algebras). As a first application we show that the Hochschild-Serre identity is the dual statement of our result. In particular, we extend this identity to Hopf algebras. Secondly, we clarify the construction of a class of quasi-Hopf algebras.     

Derived equivalences and Serre duality for Gorenstein algebras

We introduce a notion of Gorenstein algebras of codimension c and demonstrate that Serre duality theory plays an essential role in the theory of derived equivalences for Gorenstein algebras.     

Quantum group actions, twisting elements, and deformations of algebras

We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for catagories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.     

The quiver of projectives in hereditary categories with Serre duality

Let k be an algebraically closed field and A a k-linear hereditary category satisfying Serre duality with no infinite radicals between the preprojective objects. If A is generated by the preprojective objects, then we show that A is derived equivalent to for a so-called strongly locally finite quiver Q. To this end, we introduce light cone distances and round trip distances on quivers which will be used to investigate sections in stable translation quivers of the form ZQ.     

The bar derived category of a curved dg algebra

Curved A_∞-algebras appear in nature as deformations of dg algebras. We develop the basic theory of curved A_∞-algebras and, in particular, curved dg algebras. We investigate their link with a suitable class of dg coalgebras via the bar construction and produce Quillen model structures on their module categories. We define the analogue of the relative derived category for a curved dg algebra.     

Skew Clifford algebras

We introduce a generalization, called a skew Clifford algebra, of a Clifford algebra, and relate these new algebras to the notion of graded skew Clifford algebra that was defined in 2010. In particular, we examine homogenizations of skew Clifford algebras, and determine which skew Clifford algebras can be homogenized to create Artin-Schelter regular algebras. Just as (classical) Clifford algebras are the Poincaré-Birkhoff-Witt (PBW) deformations of exterior algebras, skew Clifford algebras are the ${\mathbb{Z}}_2$-graded PBW deformations of quantum exterior algebras. We also determine the possible dimensions of skew Clifford algebras and provide several examples.     

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A_Γ corresponding to an arbitrary Brauer tree Γ of type (m,e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A_S, whose tree is a star with the exceptional vertex in the middle, to A_Γ. The grading on A_S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute , the group of outer automorphisms that fix the isomorphism classes of simple A_Γ-modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A_Γ up to graded Morita equivalence and rescaling.     

From iterated tilted algebras to cluster-tilted algebras

In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.     

The Grothendieck group of a cluster category

For the cluster category of a hereditary or a canonical algebra, or equivalently for the cluster category of the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an admissible triangulated structure.     

Ext-projectives in suspended subcategories

Let H be a hereditary abelian k-category with tilting object and denote the bounded derived category of H. This paper is devoted to a study of suspended subcategories of by means of their Ext-projectives.     

On the (non)vanishing of some “derived” categories of curved dg algebras

Since curved dg algebras, and modules over them, have differentials whose square is not zero, these objects have no cohomology, and there is no classical derived category. For different purposes, different notions of “derived” categories have been introduced in the literature. In this article, we show that for some concrete curved dg algebras, these derived categories vanish. This happens for example for the initial curved dg algebra whose module category is the category of precomplexes, and for certain deformations of dg algebras.