On minimal disjoint degenerations of modules over tame path algebras

For representations of tame quivers the degenerations are controlled by the dimensions of various homomorphism spaces. Furthermore, there is no proper degeneration to an indecomposable. Therefore, up to common direct summands, any minimal degeneration from M to N is induced by a short exact sequence 0→U→M→V→0 with indecomposable ends that add up to N. We study these ‘building blocs’ of degenerations and we prove that the codimensions are bounded by two. Therefore, a quiver is Dynkin resp. Euclidean resp. wild iff the codimension of the building blocs is one resp. bounded by two resp. unbounded. We explain also that for tame quivers the complete classification of all the building blocs is a finite problem that can be solved with the help of a computer.     

Minimal singularities for representations of Dynkin quivers

We develop some reduction techniques for the study of singularities in orbit closures of finite dimensional modules. This enables us to classify all singularities occurring in minimal degenerations of representations of Dynkin quivers. They are all smoothly equivalent to the singularity at the zero-matrix inside thep×q-matrices of rank at most one.     

Quivers of finite mutation type and skew-symmetric matrices

Quivers of finite mutation type are certain directed graphs that first arised in Fomin-Zelevinsky’s theory of cluster algebras. It has been observed that these quivers are also closely related with different areas of mathematics. In fact, main examples of finite mutation type quivers are the quivers associated with triangulations of surfaces. In this paper, we study structural properties of finite mutation type quivers in relation with the corresponding skew-symmetric matrices. We obtain a characterization of finite mutation type quivers that are associated with triangulations of surfaces and give a new numerical invariant for their mutation classes.     

Auslander-Reiten quivers of local,orders of finite lattice type

The classification of the Auslander-Reiten quivers of the local orders of finite lattice type is completed. For this purpose, it is shown—using the results of [7]—that to the list of the known Auslander-Reiten quivers of the local Bäckström orders of finite lattice type [11], [14] and of the local Gorenstein orders of finite type and their minimal overorders [18] one has to add two remaining types of valued translation quivers to obtain a complete list of all Auslander-Reiten quivers of the local orders of finite lattice type.     

Graded Calabi Yau algebras of dimension 3

In this paper, we prove that Graded Calabi Yau algebras of dimension 3 are isomorphic to path algebras of quivers with relations derived from a superpotential. We show that for a given quiver Q and a degree d, the set of good superpotentials of degree d, i.e. those that give rise to Calabi Yau algebras, is either empty or almost everything (in the measure theoretic sense). We also give some constraints on the structure of quivers that allow good superpotentials, and for the simplest quivers we give a complete list of the degrees for which good superpotentials exist.     

Representation theory of Dynkin quivers. Three contributions

The representations of the Dynkin quivers and the corresponding Euclidean quivers are treated in many books. These notes provide three building blocks for dealing with representations of Dynkin (and Euclidean) quivers. They should be helpful as part of a direct approach to study represen-tations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.     

Hall Polynomials for Affine Quivers of Type Am（m≥1）

It is well known that Hall polynomials as structural coefficients play an important role in the structure of Lie algebras and quantum groups. By using the properties of representation categories of affine quivers, the task of computing Hall polynomials for affine quivers can be reduced to counting the numbers of solutions of some matrix equations. This method has been applied to obtain Hall polynomials for indecomposable representations of quivers of type Am（m≥1）     

Quiver Quotient Varieties and Complete Intersections

In this paper, we classify all the symmetric quivers and corresponding dimension vectors whose quotient space, classifying the semisimple representation classes, is a complete intersection. The result we obtain is that such quivers can be reduced to a few number of basic quivers, using some elementary types of reduction. Presented by L. Le BruynMathematics Subject Classification (2000) 16G20.     

Orthoscalar quiver representations corresponding to extended Dynkin graphs in the category of Hilbert spaces

It is known that finitely representable quivers correspond to Dynkin graphs and tame quivers correspond to extended Dynkin graphs. In an earlier paper, the authors generalized some of these results to locally scalar (later renamed to orthoscalar) quiver representations in Hilbert spaces; in particular, an analog of the Gabriel theorem was proved. In this paper, we study the relationships between indecomposable representations in the category of orthoscalar representations and indecomposable representations in the category of all quiver representations. For the quivers corresponding to extended Dynkin graphs, the indecomposable orthoscalar representations are classified up to unitary equivalence.     

On rigid quivers

We consider quivers that appear in the theory of tiled orders, in particular, rigid quivers. We prove that a quiver having a loop at each vertex is not rigid, and the quiver associated with a finite partially ordered set having one minimal element is rigid. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 105–120, 2006.     

Hopf箭图与Hopf代数

本文研究了简单无向Hopf箭图的图论性质以及路代数与路余代数的关系.利用简单无向Hopf箭图与简单图的关系.证明了路代数是路余代数的对偶的直和部分.     

Quiver representations of maximal rank type and an application to representations of a quiver with three vertices

We introduce the notion of ‘maximal rank type’ forrepresentations of quivers, which requires certain collectionsof maps involved in the representation to be of maximal rank.We show that real root representations of quivers are of maximalrank type. By using the maximal rank type property and universalextension functors we construct all real root representationsof a particular wild quiver with three vertices. From this constructionit follows that real root representations of this quiver aretree modules. Moreover, formulae given by Ringel can be appliedto compute the dimension of the endomorphism ring of a givenreal root representation.     

Auslander-Reiten theory over topological spaces

Auslander-Reiten triangles and quivers are introduced into algebraic topology. It is proved that the existence of Auslander-Reiten triangles characterizes Poincaré duality spaces, and that the Auslander-Reiten quiver is a weak homotopy invariant. The theory is applied to spheres whose Auslander-Reiten triangles and quivers are computed. The Auslander-Reiten quiver over the $d$-dimensional sphere turns out to consist of $d-1$ copies of ${\mathbb Z} A_{\infty}$. Hence the quiver is a sufficiently sensitive invariant to tell spheres of different dimension apart.     

The Algebras of Semi-Invariants of Euclidean Quivers

We give a new short proof of Skowroński and Weyman's theorem about the structure of the algebras of semi-invariants of Euclidean quivers, in the case of quivers without oriented cycles and in characteristic zero. Our proof is based essentially on Derksen and Weyman's result about the generators of these algebras and properties of Schofield semi-invariants.     

Coloured quiver mutation for higher cluster categories

We define mutation on coloured quivers associated to tilting objects in higher cluster categories. We show that this operation is compatible with the mutation operation on the tilting objects. This gives a combinatorial approach to tilting in higher cluster categories and especially an algorithm to determine the Gabriel quivers of tilting objects in such categories.     

Quantized Chebyshev polynomials and cluster characters with coefficients

We introduce quantized Chebyshev polynomials as deformations of generalized Chebyshev polynomials previously introduced by the author in the context of acyclic coefficient-free cluster algebras. We prove that these quantized polynomials arise in cluster algebras with principal coefficients associated to acyclic quivers of infinite representation types and equioriented Dynkin quivers of type \mathbbA\mathbb{A} . We also study their interactions with bases and especially canonically positive bases in affine cluster algebras.     

Quivers of semi-maximal rings

In this paper, the set of quivers of semi-maximal rings is investigated. It is proved that the elements of this set are formed by the elements of the set of quivers of tiled orders and that the set of quivers of tiled orders with n vertices is determined by the integer points of a convex polyhedral domain that lie in the nonnegative part of the space . It is also proved that the set of quivers of tiled orders with n vertices contains all simple, oriented, strongly connected graphs with n vertices and n loops, does not contain any graphs with n vertices and n − 1 loops, and contains only a part of the graphs with n vertices and m (m < n − 1) loops. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 215–223, 2005.