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)).

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）

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.

Quiver的正合表示

给定一个quiver Q和正合范畴A.证明了Q的A-表示范畴Rep(Q,A)继承了A的正合结构;当Q是左rooted quiver时确定了Rep(Q,A)的投射对象.

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.

TheIndecomposable Modules of Self InjectiveAlgebra and Their Composition Factors

In this paper, a sufficient and necessary condition for the indecomposable modules of the self injective algebra of finite representation type to be determined by their composition factors is presented by means of the quiver with relations with the help of covering.

Introduction of the Coulomb dressed potential into atomic scattering

The model of the Coulomb dressed potential is applied to solving the problem of electron scattering for simplifying the calculation in the electrostatic field and the excimer laser field. The introduction and the application of the model are based on the electric dipole approximation, so the contribution of the electric multipole is neglected. In this paper, rigorous analysis and deduction are carried out for the introduction of the dressed Coulomb potential into the laser field. It is found that the introduction of the dressed potential in the fractional form is feasible only when the laser field （not including far ultraviolet field and x-ray） is a weak field, i.e. the quiver radius of the free electron is smaller than the atomic scale. In addition, the necessary analysis is also conducted of the limitation of the application of the Coulomb dressed potential.

On the structure and representations of non-balanced quantum doubles

For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space （kX°P）＊ kG （see Theorem 2.1）, called a non-balanced quantum double and denoted by Dx（G）. In this paper, some Hopf algebra properties of Dx （G） are given, the representation types of Dx （G） viewed as a k-algebra are discussed, the algebra structure and module category over Dx（G） are studied. Since the Hopf algebra structure of non-balanced quantum double DX （G） generMizes the usual quantum double D（G） for a finite group G, all results about Dx（G） in this paper can also be used to describe D（G） as a special case and the universal R-matrix of Dx （G） provides more solutions of Yang-Baxter equation.

Two Dimensional Indecomposable Modules over Infinite Dimensional Hereditary Path Algebras

Non-isomorphic two dimensional indecomposable modules over infinite dimensional hereditary path algebras are described. We infer that none of them can be determined by their dimension vectors.

On the McKay Quiver

Using the connection between McKay quiver and Loewy matrix, and the properties of characteristic polynomial of Loewy matrix, we give a generalized way to determine the McKay quiver for a finite subgroup of a generalized linear group.

一门多学,一课多元——基于Quiver App的电脑绘画拓展性课程的实施研究

大数据时代下,移动终端融入教学已经成一种趋势,高科技给生活带来的方便大家都有目共睹,那么在教育方面的应用,教育者们也一直都在努力探索。文章是以移动终端融入教学为前提,结合Quiver App应用与电脑绘画之间的关联展开,通过对课程的整合,进行多元化融合,体现一门多学、一课多元的特点。