Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.     

On tame weakly symmetric algebras having only periodic modules

We classify (up to Morita equivalence) all tame weakly symmetric finite dimensional algebras over an algebraically closed field having simply connected Galois coverings, nonsingular Cartan matrices and the stable Auslander-Reiten quivers consisting only of tubes. In particular, we prove that these algebras have at most four simple modules.Received: 25 February 2002     

The classical Galois closure for universal algebras

We study the Galois correspondence between subgroups of groups of universal algebras automorphisms and subalgebras of fixed points of these automorphisms.     

On the Galois coverings of a cluster-tilted algebra

We study the module category of a certain Galois covering of a cluster-tilted algebra which we call the cluster repetitive algebra. Our main result compares the module categories of the cluster repetitive algebra of a tilted algebra C and the repetitive algebra of C, in the sense of Hughes and Waschbüsch.     

Quivers with Relations and Cluster Tilted Algebras

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.Presented by V. Dlab.     

Stable Equivalence of Selfinjective Artin Algebras of Dynkin Type

We prove new results on the stable equivalences of selfinjective Artin algebras of tilted Dynkin type, extending the main results of Skowroński and Yamagata to arbitrary tilted type.Presented by I. Reiten     

Nakayama oriented pullbacks and stably hereditary algebras

We introduce a new class of algebras, the Nakayama oriented pullbacks, obtained from pullbacks of surjective morphisms of algebras A?C and B?C. We prove that such a pullback is tilted when A and B are hereditary. We also show that stably hereditary algebras respecting the clock condition are Nakayama oriented pullbacks, and we use results about these pullbacks to show when a stably hereditary algebra is tilted or iterated tilted.     

The number of the Gabriel-Roiter measures admitting no direct predecessors over a wild quiver

A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel-Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary Artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel-Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of Artin algebras. As the first step, we are going to study the wild quiver with three vertices, labeled by 1, 2 and 3, and one arrow from 1 to 2 and two arrows from 2 to 3. The Gabriel-Roiter submodules of the indecomposable preprojective modules and quasi-simple modules τ⁻ⁱM, i≥0 are described, where M is a Kronecker module and τ=DTr is the Auslander-Reiten translation. Based on these calculations, the existence of infinitely many GR segments will be shown. Moreover, it will be proved that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors.     

Selfinjective algebras without short cycles of indecomposable modules

We describe the structure of finite dimensional selfinjective algebras over an arbitrary field without short cycles of indecomposable modules.     

Descent constructions for central extensions of infinite dimensional Lie algebras

We use Galois descent to construct central extensions of twisted forms of split simple Lie algebras over rings. These types of algebras arise naturally in the construction of Extended Affine Lie Algebras. The construction also gives information about the structure of the group of automorphisms of such algebras. A. Pianzola is supported by the NSERC Discovery Grant Program. The author also wishes to thank the Instituto Argentino de Matemática for their hospitality. D. Prelat is supported by a Research Grant from Universidad CAECE.     

Rhombus filtrations and Rauzy algebras

Peach introduced rhombal algebras associated to quivers given by tilings of the plane by rhombi. We develop general techniques to analyze rhombal algebras, including a filtration by what we call rhombus modules. We introduce a way to relate the infinite-dimensional rhombal algebra corresponding to a complete tiling of the plane to finite-dimensional algebras corresponding to finite portions of the tiling. Throughout, we apply our general techniques to the special case of the Rauzy tiling, which is built in stages reflecting an underlying self-similarity. Exploiting this self-similar structure allows us to uncover interesting features of the associated finite-dimensional algebras, including some of the tree classes in the stable Auslander-Reiten quiver.     

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).     

Some characterisations of supported algebras

We give several equivalent characterisations of left (and hence, by duality, also of right) supported algebras. These characterisations are in terms of properties of the left and the right parts of the module category, or in terms of the classes L₀ and R₀ which consist respectively of the predecessors of the projective modules, and of the successors of the injective modules.     

On Lie algebras associated with representation-directed algebras

Let B be a representation-finite C-algebra. The Z-Lie algebra L(B) associated with B has been defined by Riedtmann in [Ch. Riedtmann, Lie algebras generated by indecomposables, J. Algebra 170 (1994) 526-546]. If B is representation-directed, there is another Z-Lie algebra associated with B defined by Ringel in [C.M. Ringel, Hall Algebras, vol. 26, Banach Center Publications, Warsaw, 1990, pp. 433-447] and denoted by K(B).We prove that the Lie algebras L(B) and K(B) are isomorphic for any representation-directed C-algebra B.     

Morita equivalence of quasi-primal algebras and sheaves

In this paper, we show that two quasi-primal algebras are Morita equivalent if and only if their inverse semigroups of inner automorphisms are isomorphic, and if they have the same one-element subalgebras. The proof of this statement uses the representation theory of algebras by sections in sheaves.Presented by H. P. Gumm.     

Semi-invariants of canonical algebras

We describe the algebras of semi-invariants on the varieties of regular representations of canonical algebras. In particular, we show that these algebras are polynomial algebras or complete intersections. Received: 29 March 1999     

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 Gabriel-Roiter measure

The first Brauer-Thrall conjecture asserts that algebras of bounded representation type have finite representation type. This conjecture was solved by Roiter in 1968. The induction scheme which he used in his proof prompted Gabriel to introduce an invariant which we propose to call Gabriel-Roiter measure. This invariant is defined for any finite length module and it will be studied in detail in this paper. Whereas Roiter and Gabriel were dealing with algebras of bounded representation type only, it is the purpose of the present paper to demonstrate the relevance of the Gabriel-Roiter measure for algebras in general, in particular for those of infinite representation type.