Cluster-tilted algebras

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

     

Isomorphic trivial extensions of finite dimensional algebras

Let Λ be a finite dimensional algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of Λ is zero in Λ. Let T(Λ)=Λ?D(Λ) be the trivial extension of Λ by its minimal injective cogenerator D(Λ). We characterize, in terms of quivers and relations, the algebras Λ^′ such that T(Λ)?T(Λ^′).     

Cluster-tilted algebras and slices

We give a criterion allowing to verify whether or not two tilted algebras have the same relation-extension (thus correspond to the same cluster-tilted algebra). This criterion is in terms of a combinatorial configuration in the Auslander–Reiten quiver of the cluster-tilted algebra, which we call local slice.     

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A×DA in terms of the combinatorics of the quiver and relations is determined.     

Examples of rings and modules as trivial extensions

Abstract

The two main classes of finite dimensional composition algebras, Hurwitz and symmetric, are characterized in terms of the flexible identity, and even of the third power associative identity, over fields of arbitrary characteristic.     

Graded self-injective algebras “are” trivial extensions

For a positively graded artin algebra A=_n0A_n we introduce its Beilinson algebra b(A). We prove that if A is well-graded self-injective, then the category of graded A-modules is equivalent to the category of graded modules over the trivial extension algebra T(b(A)). Consequently, there is a full exact embedding from the bounded derived category of b(A) into the stable category of graded modules over A; it is an equivalence if and only if the 0-th component algebra A₀ has finite global dimension.     

Tilting and trivial extensions

The trivial extensions of a quasi-abelian category by means of a fully exact endofunctor are again quasi-abelian. Using the one-to-one correspondence between quasi-abelian categories and tiltings, it is shown that the trivial extension of a quasi-abelian category corresponds to a trivial extension of the associated tilting. As an application, a criterion for tilting modules over an arbitrary ring R to be liftable to a tilting module over a trivial extension ring R\ltimes M{R\ltimes M} is given.     

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.     

Automorphism groups of trivial extensions

Given a split basic finite dimensional algebra A over a field, we study the relationship between the groups of categorical automorphisms of A and its trivial extension A?D(A). Our results cover all triangular algebras and all 2-nilpotent algebras whose quiver has no nontrivial oriented cycle of length ?2. In this latter as well as in the hereditary case, we give structure theorem for CAut(A?D(A)) in terms of CAut(A). As a byproduct, we get the precise relationship between the first Hochschild cohomology groups of A and A?D(A).     

Representation-finite trivial extension algebras

Let A be a finite-dimensional basic connected associative algebra over an algebraically closed field, and $\text{T(A)=A ? DA}$ its trivial extension by its minimal injective cogenerator. We prove that T(A) is representation-finite of Cartan class Δ if and only if A is an iterated tilted algebra of Dynkin class Δ. The proof also yields a construction procedure for iterated tilted algebras of Dynkin type.     

Slightly trivial extensions of a fusion category

We introduce and study the notion of slightly trivial extensions of a fusion category which can be viewed as the first level of complexity of extensions. We also provide two examples of slightly trivial extensions which arise from rank 3 fusion categories.     

On components of the Auslander-Reiten quiver of trivial extensions of 2-fundamental algebras which contain projective modules

Trivial extensions of a certain subclass of minimal 2-fundamental algebras are examined. For such algebras the characterization of components of the Auslander-Reiten quiver which contain indecomposable projective modules is given.