José Antonio de la Peña

The fundamental groups of a triangular algebra

A finite dimensional algebra A (over an algebraically closed field) is called triangular if its ordinary quiver has no oriented cycles. To each presentation (Q I) of A is attached a fundamental group π1(Q I), and A is called simply connected if π1(Q I) is trivial for every presentation of A. In this paper, we provide tools for computations with the fundamental groups, as well as criteria for simple connectedness. We find relations between the fundamental groups of A and the first Hochschild cohomology H 1 (A A).

CONCEALED-CANONICAL ALGEBRAS AND SEPARATING TUBULAR FAMILIES

We characterise those concealed-canonical algebras which arise as endomorphism rings of tilting modules, all of whose indecomposable summands have strictly positive rank, as those artin algebras whose module categories have a separating exact subcategory (that is, a separating tubular family of standard tubes). This paper develops further the technique of shift automorphisms which arises from the tubular structure. It is related to the characterisation of hereditary noetherian categories with a tilting object as the categories of coherent sheaves on a weighted projective line.

Finite cycles of indecomposable modules

This work was completed with the support of the research grant DEC-2011/02/A/ST1/00216 of the Polish National Science Center and the CIMAT Guanajuato, Mexico.

On the number of terms in the middle of almost split sequences over cycle-finite artin algebras

We prove that the number of terms in the middle of an almost split sequence in the module category of a cycle-finite artin algebra is bounded by 5.

Cycle-Finite Module Categories

We describe the structure of module categories of finite dimensional algebras over an algebraically closed field for which the cycles of nonzero nonisomorphisms between indecomposable finite dimensional modules are finite (do not belong to the infinite Jacobson radical of the module category). Moreover, geometric and homological properties of these module categories are exhibited.

ALGEBRAS WITH CYCLE-FINITE GALOIS COVERINGS

We prove that the finite dimensional algebras over an algebraically closed field which admit cycle-finite Galois coverings with torsion-free Galois groups are of tame representation type, and derive some consequences.

Tilt-Critical Algebras of Tame Type

Let A be a finite dimensional k-algebra over an algebraically closed field. Assume A = kQ/I where Q is a quiver without oriented cycles. We say that A is tilt-critical if it is not tilted but every proper convex subcategory of A is tilted. We describe the tilt-critical algebras which are strongly simply connected and tame.

Stable representations of quivers

Let Q be a finite quiver without oriented cycles and let kQ the path algebra of Q over an algebraically closed field k. We investigate stable finite dimensional representations of Q. That is for a fixed dimension vector d and a fixed weight θ we consider θ-stable representations of Q with dimension vector d. If we wish to compare also representations with different dimension vectors, then it is more convenient to consider a slope μ instead of a weight θ. In particular, we apply the results of Harder–Narasimhan on natural filtrations associated to any fixed slope μ to the category of representations of Q. Further we introduce the wall system for weights with respect to a fixed dimension vector d and consider several examples.

Algebras whose Tits form accepts a maximal omnipresent root

Let k be an algebraically closed field and A be a finite-dimensional associative basic k-algebra of the form A=kQ/I where Q is a quiver without oriented cycles or double arrows and I is an admissible ideal of kQ. We consider roots of the Tits form q_A, in particular in case q_A is weakly non-negative. We prove that for any maximal omnipresent root v of q_A, there exists an indecomposable A-module X such that v is the dimension vector of X. Moreover, if A is strongly simply connected, the existence of a maximal omnipresent root of q_A implies that A is tame of tilted type.

On the inductive construction of Galois coverings of algebras

Let A be a finite-dimensional algebra over an algebraically closed field k. In order to study the category modA of finitely generated left A-modules, we may assume that A is basic and connected. By [8], there is a finite quiver (i.e. oriented graph) QA and a surjective homomorphism of algebras ν : kQA → A such that Iν = kerν ⊂ (kQA)2, where kQA denotes the path algebra associated to QA and kQ + A denotes the ideal of kQA generated by all arrows in QA. For each pair (QA, Iν), called a presentation of A, we can define the fundamental group π1(QA, Iν) (see [9,13] or Section 1.2 below). Then there is surjective homomorphism of algebras F : Aν →A defined by the action of π1(QA, Iν) on Aν , called the universal Galois covering of A with respect to ν. As in [8], we shall consider algebras as locally finite k-categories. Galois coverings have proved to be a powerful tool for the study of the module category modA. Indeed, A is representation-finite if and only if Aν is locally support-finite and locally representation-finite; in this case, if chark = 2, then Aν = kQ/Ĩ where Q is a quiver without oriented cycles [4,9,14]. If A is tame, then A is tame [6] but the converse does not hold [11]. Certain A-modules may be described via the push-down functor Fλ : mod Aν → modA, see [5]. In this paper we shall consider only triangular algebras, that is, algebras A= kQA/Iν such that QA has no oriented cycles. In this case, a vertex x in QA is said to be separating if for the indecomposable decomposition radPx = M1 ⊕ · · · ⊕ Ms , there is a decomposition into connected components Q A = Q1 ∐ . . . ∐ Qt , with t s, of the induced full subquiver Q A of QA with vertices those y which are not predecessors