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.     

A functor between categories of regular modules for wild hereditary algebras

Supported by the SERC of Great Britain     

The tame dimension vectors for trees

We consider representations of quivers over an algebraically closed field K. A dimension vector of a quiver is called hypercritical, if there is an m-parameter family of indecomposable representations for the dimension vector with m?2, but every family of representations for all smaller dimension vectors depends on a single parameter. We characterise the hypercritical dimension vectors for trees via their Tits forms and those of their decompositions and present the complete list of the hypercritical dimension vectors.Finally, this leads to a combinatorial classification of the tame dimension vectors for trees which is also given by the Tits forms.     

The tame dimension vectors for stars

We consider representations of stars over an algebraically closed field K. We classify those dimension vectors of stars admitting a one parameter family of indecomposable representations and for which, in addition, all families of (not necessarily indecomposable) representations depend on a single parameter. Furthermore, we show how it is possible to construct the corresponding one parameter families of indecomposable representations.     

Submodules of Kronecker modules via extension monoid products

Using some results on Reineke's extension monoid product over the Kronecker algebra kQ, we provide numerical criteria in terms of Kronecker invariants for a module in $\mathrm{mod}\text{-}kQ$ to be isomorphic with the submodule of an another module in $\mathrm{mod}\text{-}kQ$. The results can be transferred to matrix pencils in solving an important challenge in matrix pencil theory: characterize in terms of Kronecker invariants when a given pencil is a subpencil of an another one.     

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.     

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.     

Linear quivers, generic extensions and Kashiwara operators

In the present paper, we introduce the generic extension graph G of a Dynkin or cyclic quiver Q and then compare this graph with the crystal graph C for the quantized enveloping algebra associated to Q via two maps ℘_Q, _Q : Ω → Λ_Q induced by generic extensions and Kashiwara operators, respectively, where Λ_Q is the set of isoclasses of nilpotent representations of Q, and Ω is the set of all words on the alphabet I, the vertex set of Q. We prove that, if Q is a (finite or infinite) linear quiver, then the intersection of the fibres ℘_Q⁻¹ (λ) and KQ⁻¹ (λ) is non-empty for every λ ∈ Λ _Q. We will also show that this non-emptyness property fails for cyclic quivers.     

On the zero set of semi-invariants for tame quivers

Let d be a prehomogeneous dimension vector for a finite tame quiver Q. We show that the common zeros of all non-constant semi-invariants for the variety of representations of Q with dimension vector $N\cdot\mathbf d$, under the product of the general linear groups at all vertices, is a complete intersection for $N\geq 3$.     

On the Kronecker Problem and related problems of Linear Algebra

We consider some classification problems of Linear Algebra related closely to the classical Kronecker Problem on pairs of linear maps between two finite-dimensional vector spaces. As shown by Djokovi? and Sergeichuk, the Kronecker’s solution is extended to the cases of pairs of semilinear maps and (more generally) pseudolinear bundles respectively. Our objective is to deal with the semilinear case of the Kronecker Problem, especially with its applications. It is given a new short solution both to this case and to its contragredient variant. The biquadratic matrix problem is investigated and reduced in the homogeneous case (in characteristic ≠2) to the semilinear Kronecker Problem. The integer matrix sequence Θ_n and Θ-transformation of polynomials are introduced and studied to get a simplified canonical form of indecomposables for the mentioned homogeneous problem. Some applications to the representation theory of posets with additional structures are presented.     

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles.     

Cleaving functors and controlled wild algebras

We show that the cleaving functors introduced in [Bautista et al., Invent. Math. 81 (1985) 217] as a tool for proving infinite representation type of finite-dimensional algebras can also be used to establish controlled wildness. The main application is that an algebra is controlled wild if there is an indecomposable projective module with a Loewy factor having a homogeneous direct summand which is of length at least 3. As a second application we derive Han's covering criterion.     

Wild cluster tilted algebras of rank 3

Let H be a connected wild hereditary algebra of rank 3, T a square-free tilting H-module and the endomorphism ring of T in the cluster category C_H of H. It is shown that the Loewy length of the cluster tilted algebra Γ is 3+r, where r denotes the number of regular indecomposable direct summands of T.     

Degrees of irreducible morphisms in standard components

We study the left degree of an irreducible morphism with X and Y_i indecomposable modules in a standard component of the Auslander-Reiten quiver, for 1≤i≤r. Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two.     

Orbit semigroups and the representation type of quivers

We show that a finite, connected quiver Q without oriented cycles is a Dynkin or Euclidean quiver if and only if all orbit semigroups of representations of Q are saturated.     

The N = 1 quivers of types An and Dn have finite representation type

We show that under certain conditions, the N = 1 types A and D quivers are of finite representation type.     

Localization in tame and wild coalgebras

We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: “Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations”.     

Large indecomposables over representation-infinite orders and algebras

Let R be a complete discrete valuation domain with quotient field K, and let be an R-order in a semisimple K-algebra. Butler, Campbell, and Kovács have shown that R-free -modules decompose into -lattices when is representation-finite. Using the theory of ladder functors, we prove the converse by constructing indecomposable R-free -modules of infinite rank if is not representation-finite.Received: 23 March 2004