Phases of Stable Representations of Euclidean and Wild Quivers

We consider stable representations of non-Dynkin quivers with respect to a central charge.These attract a lot of interest in mathematics and physics since they can be identified with so-called BPS states.Another motivation is the work of Dimitrov et al.on the phases of stable representations of the generalized Kronecker quiver.One aim is to explain for general Euclidean and wild quivers the behavior of phases of stable representations well known in some examples.In addition,we study especially the behavior of preinjective,postprojective and regular indecomposable modules.We show that the existence of a stable representation with self-extensions implies the existence of infinitely many stables without self-extensions for rigid central charges.In this case the phases of the stable representations approach one or two limit points.In particular,the phases are not dense in two arcs.The category of representations of acyclic quivers is a special case of rigid Abelian categories which show this behavior for rigid central charges.

Completeness of Induced Cotorsion Pairs in Representation Categories of Rooted Quivers

This paper focuses on a question raised by Holm and Jorgensen,who asked if the induced cotorsion pairs(Φ(X),Φ(X)^(⊥))and(^(⊥)Ψ(Y),Ψ(Y))in Rep(Q,A)—the category of all A-valued representations of a quiver Q—are complete whenever(X,Y)is a complete cotorsion pair in an abelian category A satisfying some mild conditions.We give an affirmative answer if the quiver Q is rooted.As an application,we show under certain mild conditions that if a subcategory L,which is not necessarily closed under direct summands,of A is special precovering(resp.,preenveloping),thenΦ(L)(resp.,Ψ(L))is special precovering(resp.,preenveloping)in Rep(Q,A).

On Sectional Cycles in Translation Quivers

In this paper, we prove that there exists no sectional cycle in a translation quiver under certain conditions. So, we generalize Dautista and Smalφ's corresponding result on AR-quiver of an artin algebra.

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 representations of quivers, and they shed some new light on properties of Dynkin and Euclidean quivers.

Construct non-graded bi-Frobenius algebras via quivers

Using the quiver technique we construct a class of non-graded bi-Frobenius algebras. We also classify a class of graded bi-Frobenius algebras via certain equations of structure coefficients.

Generalized McKay Quivers of Rank Three

For each finite subgroup G of SLn（C）, we introduce the generalized Cartan matrix AG in view of McKay correspondence from the fusion rule of its natural representation. Using group theory, we show that the generalized Cartan matrices have similar favorable properties such as positive semi- definiteness as in the classical case of affine Cartan matrices. The complete McKay quivers for SL3 （C） are explicitly described and classified based on representation theory.

On the McKay quivers and m-Cartan matrices Dedicated to Professor LIU ShaoXue on the occasion of his 80th birthday

In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group.

AR quivers,exceptional sequences and algorithms in derived Hall algebras

Consider the canonical isomorphism between the positive part U+ of the quantum group Uq(g) and the Hall algebra H(Λ),where the semisimple Lie algebra g and the finite-dimensional hereditary algebra Λ share a Dynkin diagram.Chen and Xiao have given two algorithms to decompose the root vectors into linear combinations of monomials of Chevalley generators of U+,respectively induced by the braid group action on the exceptional sequences of Λ-modules and the structure of the Auslander-Reiten quiver of Λ.In this paper,we obtain the corresponding algorithms for the derived Hall algebra DH(Λ),which was introduced by Toen.We show that both algorithms are applicable to the lattice algebra and Heisenberg double in the sense of Kapranov.All the new recursive formulae have the same flavor with the quantum Serre relations.

Semilinear automorphisms of classical groups and quivers

For a classical group G over a field F together with a finite-order automorphism θ that acts compatibly on F, we describe the fixed point subgroup of θ on G and the eigenspaces of θ on the Lie algebra g in terms of cyclic quivers with involution. More precise classification is given when g is a loop Lie algebra, i.e.,when F = C((t)).

McKay quivers and absolute n-complete algebras

In this paper,we show that the McKay quiver of a finite subgroup of a general linear group is a regular covering of the McKay quiver of its intersection with the special linear group.Using this and our results on "returning arrows" in McKay quiver,we give an algorithm to construct the McKay quiver of a finite abelian group.Using this construction,we show how the cone and cylinder of an(n-1)-Auslander absolute n-complete algebra are truncated from the McKay quivers of abelian groups.

Homologies of Infinite Quivers

For a path algebra A = kQ with Q an arbitrary quiver, consider the Hochschild homology groups Hn(A) and the homology groups TornAe(A, A), where Ae is the enveloping algebra of A. In this paper the groups are explicitly given.

On duality preservability of Auslander-Reiten quivers of derived categories and cluster categories

We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. of generalized path algebras as cleft Meantime, we characterize the condition extensions of path algebras.

Hall Polynomials for Affine Quivers of Type _m(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）

Semistable representations of quivers with automorphism

Let Q be a quiver with automorphism σ. We prove that semistable representations of Q over F_q give rise to semistable modules over the F_q-algebra A(Q, σ; q) associated with(Q, σ). As an application, we obtain a description of the semistable subcategories of A(Q, σ; q)-modules and determine the slopes of semistable A(Q, σ; q)-modules in the case that Q is a Dynkin or tame quiver.

Cluster characters for cyclic quivers

We define an analogue of the Caldero-Chapoton map for the cluster category of finite-dimensional nilpotent representations over a cyclic quiver. We prove that it is a cluster character and satisfies some inductive formulas for the multiplication between the generalized cluster variables （the images of objects of the cluster category under this map）. Moreover, we construct a Z-basis for the algebra generated by all generalized cluster variables.

F-theory and N=1 Quivers from Polyvalent Geometry

We study four-dimensional quiver gauge models from F-theory compactified on fourfolds with hyper-K¨ahler structure.Using intersecting complex toric surfaces,we derive a class of N =1 quivers with charged fundamental matter placed on external nodes.The emphasis is on how local Calabi–Yau equations solve the corresponding physical constraints including the anomaly cancelation condition.Concretely,a linear chain of SU(N) groups with flavor symmetries has been constructed using polyvalent toric geometry.

Biserial Incidence Algebras

Let K be an algebraically closed field and A be a finite dimensional algebra over K. In this paper we give a classification of biserial incidence algebras with quiver methods.

Homological Properties in Relation to Nakayama Algebras

The goal of this paper is to investigate whether the Ext-groups of all pairs (M, N) of modules over Nakayama algebras of type (n,n,n) satisfy the condition ExtΛn(M,N)=0 for n >> 0 ? ExtΛn(M,N)=0 for n >> 0. We achieve that by discussing the Ext-groups of Nakayama algebra with projectives of lengths 3n+1 and 3n+2 using combinations of modules of different lengths.

用quiver构造拟三角Hopf代数

利用quiver方法确定了一个广义Taft代数具有拟三角Hopf结构当且仅当它是Sweedler 4维Hopf代数．用不同于文[15]的方法,对任意的正整数n,构造出一类拟三角Hopf代数H(n)．

Conservation of Quiver Trees in Namibia and South Africa under a Changing Climate

Aloe dichotoma (Quiver tree) occurs in the arid regions of Namaqualand and Bushman land in South Africa, and in arid regions of southern Namibia. The Quiver trees are not only threatened by agricultural expansion, overgrazing, and mining;but also by climate changes and droughts. Previous studies show that Quiver trees are very sensitive to environmental changes, and do not respond well to extreme hot and dry conditions. This study investigates the current status of the Quiver tree within its existing environment, and also assesses the projected future changes of the Quiver tree habitat under different climatic scenarios. It provided evidence regarding the importance of the study to understanding the climate change impacts on the Quiver tree and its geographical response to climate changes.