Presenting degenerate Ringel-Hall algebras of cyclic quivers

Using the generators labelled by simple and sincere semisimple modules for the Ringel-Hall algebra H_q(n) of a cyclic quiver Δ(n), we give a presentation for the degenerate algebra H₀(n). This is achieved by establishing a presentation for the generic extension monoid algebra of Δ(n). As an application, we show that both the degenerate Ringel-Hall algebra and the degenerate quantum affine sl_n admit multiplicative bases.     

Admissible sequences and the preprojective component of a quiver

This paper concerns indecomposable preprojective modules over the path algebra of a finite connected quiver without oriented cycles. For each such module, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the module. An efficient way to compute the module is to recover it from its shortest (+)-admissible sequence. The set of equivalence classes of the above sequences has a natural structure of a partially ordered set. For a large class of quivers, the Hasse diagram of the partially ordered set is isomorphic to the preprojective component of the Auslander-Reiten quiver. The techniques of (+)-admissible sequences yield a new result about slices in the preprojective component.     

Sequences of reflection functors and the preprojective component of a valued quiver

This paper concerns preprojective representations of a finite connected valued quiver without oriented cycles. For each such representation, an explicit formula in terms of the geometry of the quiver gives a unique, up to a certain equivalence, shortest (+)-admissible sequence such that the corresponding composition of reflection functors annihilates the representation. The set of equivalence classes of the above sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver. The results apply to the study of reduced words in the Weyl group associated with an indecomposable symmetrizable generalized Cartan matrix.     

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 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.     

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.     

Free groups and involutions in the unit group of a group algebra

We study the existence of free groups of rank 2 in the group generated by the involutions in the unit group of a group algebra over a non-absolute field.Received: 25 March 2004     

Generalized tilting functors and Ringel-Hall algebras

It is known from [M. Auslander, M.I. Platzeck, I. Reiten, Coxeter functors without diagrams, Trans. Amer. Math. Soc. 250 (1979) 1-46] and [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] that the Bernstein-Gelfand-Ponomarev reflection functors are special cases of tilting functors and these reflection functors induce isomorphisms between certain subalgebras of Ringel-Hall algebras. In [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] the result from [C.M. Ringel, PBW-basis of quantum groups, J. Reine Angew. Math. 470 (1996) 51-85] is generalized to the tilting module case by giving an isomorphism between two Ringel-Hall subalgebras. In [J. Miyashita, Tilting Modules of Finite Projective Dimension, Math. Z. 193 (1986) 113-146] Miyashita generalized the tilting theory by introducing the tilting modules of finite projective dimension. In this paper the result in [A. Wufu, Tilting functors and Ringel-Hall algebras, Comm. Algebra 33 (1) (2005) 343-348] is generalized to the tilting modules of finite projective dimension.     

Translation quiver varieties

We introduce a framework of translation quiver varieties which includes Nakajima quiver varieties as well as their graded and cyclic versions. An important feature of translation quiver varieties is that the sets of their fixed points under toric actions can be again realized as translation quiver varieties. This allows one to simplify quiver varieties in several steps. We prove that translation quiver varieties are smooth, pure and have Tate motivic classes. We also describe an algorithm to compute those motivic classes.     

Radical embeddings and representation dimension

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.     

Cyclic homology of truncated quiver algebras

We determine the cyclic homology of truncated quiver algebras over a general commutative unital ring. Thus we improve and generalise the results in [8] and [5], where this problem has been studied for algebras over a field. We use the spectral sequence of a double complex that is obtained from minimal projective bimodule resolution and comparison morphisms.     

Tame concealed algebras and cluster quivers of minimal infinite type

The well-known list of Happel-Vossieck of tame concealed algebras in terms of quivers with relations, and the list of A. Seven of minimal infinite cluster quivers are compared. There is a 1-1 correspondence between the items in these lists, and we explain how an item in one list naturally corresponds to an item in the other list. A central tool for understanding this correspondence is the theory of cluster-tilted algebras.     

Tensor products on quiver representations

The aim of this article is to translate the well-known tensor product of representations of a group given by diagonal action to the case of representations of a quiver. We provide three different approaches and exhibit their close relationship to the point-wise tensor product, which is considered in [M. Herschend, Solution to the Clebsch-Gordan problem for Kronecker representations. U.U.D.M Project Report 2003:P1, Uppsala University, 2003; M. Herschend, Solution to the Clebsch-Gordan problem for representations of quivers of type , J. Algebra Appl. 4 (5) (2005) 481-488; M. Herschend, On the Clebsch-Gordan problem for quiver representations. U.U.D.M Report 2005:43, Uppsala University, 2005].     

The rational group algebra of a normally monomial group

In this paper, a complete irredundant set of a class of strong Shoda pairs of a finite group G is computed. The algebraic structure of the rational group algebra of a normally monomial group is thus obtained. A necessary and sufficient condition for G to be normally monomial is derived. The main result is also illustrated by computing a complete set of primitive central idempotents and the explicit Wedderburn decomposition of the rational group algebra of some normally monomial groups.     

On the number of stable quiver representations over finite fields

We prove a new formula for the generating function of polynomials counting absolutely stable representations of quivers over finite fields. The case of irreducible representations is studied in more detail.     

Symmetric designs and geometroids

A-setS in a symmetric 2-(v, k, ) design is a subset which every block meets in 0, 1 or points such that for any point ofS there is a unique block meetingS at that point only. Ovoids in three-dimensional projective spaces are examples of-secs. It is shown that if has a-set then is a geometroid withv=u ²+u+1 andk=u+1, whereu–1. The cases whenu is–1, and+1 are investigated and some open problems discussed.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000