Quivers of Associative Rings

The concept of a quiver, i.e., a direct graph of a finite-dimensional algebra, was introduced by Gabriel in connection with problems of representation theory of such algebras. The notion of a scheme (=quiver Q(A)) of a semiperfect, right Noetherian ring A was introduced by the author and was applied to the study of the structure of serial rings. We give a short review of some results on structural ring theory, which were obtained by using the graph technique. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 14, Algebra, 2004.     

Unitarizable Representations of Quivers

We investigate representations of *-algebras associated with posets. Unitarizable representations of the corresponding (bound) quivers (which are polystable representations for some appropriately chosen slope function) give rise to representations of these algebras. Considering posets which correspond to unbound quivers this leads to an ADE-classification which describes the unitarization behaviour of their representations. Considering posets which correspond to bound quivers, it is possible to construct unitarizable representations starting with polystable representations of related unbound quivers which can be glued together with a suitable direct sum of simple representations. Finally, we estimate the number of complex parameters parametrizing irreducible unitary non-equivalent representations of the corresponding algebras.     

Self-Dual Hopf Quivers

ABSTRACT

We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004 van Oystaeyen , F. , Zhang , P. ( 2004 ). Quiver Hopf algebras . J. Algebra 280 ( 2 ): 577 – 589 . [CSA] [CROSSREF]  [Google Scholar]). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000 Andruskiewitsch , N. , Schneider , H.-J . ( 2000 ). Finite quantum groups and Cartan matrices . Adv. Math. 154 : 1 – 45 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.     

无限箭图的同调群

对任意箭图Q,我们研究路代数A=kQ的Hochschild同调群H_n(A)和同调群Tor_n^Ae(A,A),其中A^e是代数A的包络代数。在本文中,我们具体地给出了各次同调群和Hochschild同调群。     

Categorical Realizations of Quivers

We introduce and study categorical realizations of quivers. This construction generalizes comma categories and includes representations of quivers on categories, twisted representations of quivers (in the sense of [9 Gothen, P. B., King, Alastair D. (2005). Homological algebra of twisted quiver bundles. J. London Math. Soc. (2) 71(1):85–99.[Crossref], [Web of Science ®] , [Google Scholar]]), and bilinear pairings as special cases. In this general context, we prove the cancelation from direct sums, and show the existence and uniqueness of decomposition into a sum of indecomposable objects, provided certain assumptions hold. This yields similar results for the special cases just mentioned. Using similar ideas, we also prove a version of Fitting's Lemma for natural transformations between functors.     

Quivers and Poisson structures

We produce natural quadratic Poisson structures on moduli spaces of representations of quivers. In particular, we study a natural Poisson structure for the generalised Kronecker quiver with 3 arrows.     

Projective Representations of Quivers #

In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A _∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A _∞, which will allow us to characterize the projective representations of A _∞. This will improve some previous results and make more accurate the statement made in Benson (1991 Benson , D. J. ( 1991 ). Representations and Cohomology I . Cambridge Studies in Advanced Mathematics , Vol. 30 . Cambridge : Cambridge University Press . [Google Scholar]). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module.     

Finitely Represented K-Marked Quivers

We present necessary and sufficient conditions for the finite representability of K-marked quivers.     

Local Quivers and Stable Representations

Abstract

In this paper we introduce and study the local quiver as a tool to investigate the étale local structure of moduli spaces of θ-semistable representations of quivers. As an application we determine the dimension vectors associated to irreducible representations of the torus knot groups G _p,q  = ?a, b ∣ a  ^p  = b ^q ?.     

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)．     

Recovering Quivers from Derived Quiver Representations

We compute Balmer's prime spectrum for the derived category of quiver representations for a finite ordered quiver with the vertex-wise tensor product and show that it does not recover the quiver. We then associate an algebra to every k-linear triangulated tensor category and show that the path algebra can be recovered in this way.     

关于平移箭图上的截点圈

在本文中,我们证明了在一定条件下平移箭图中不存在截点圈(sectional cycle),从而推广了在阿丁代数的AR-箭图上Bautista和Smalφ的相应结果。     

Strong Exceptional Sequences Provided by Quivers

Let Q be a finite quiver without oriented cycles. Denote by U (Q) the fine moduli space of stable thin sincere representations of Q with respect to the canonical stability notion. We prove Ext _(Q) ^l (U, U) = 0 for all l > 0 and compute the endomorphism algebra of the universal bundle U. Moreover, we obtain a necessary and sufficient condition for when this algebra is isomorphic to the path algebra of the quiver Q. If so, then the bounded derived categories of finitely generated right k Q-modules and that of coherent sheaves on (Q) are related via the full and faithful functor – _kQ ^L U.     

Hopf Algebras on Schurian Quivers

Let H be a Hopf algebra with a finite-dimensional, nontrivial space of skew primitive elements, over an algebraically closed field of characteristic zero. We prove that H contains either the polynomial algebra as a Hopf subalgebra, or a certain Schurian simple-pointed Hopf subalgebra. As a consequence, a complete list of the locally finite, simple-pointed Hopf algebras is obtained. Also, the graded automorphism group of a Hopf algebra on a Schurian Hopf quiver is determined, and the relation between this group and the automorphism groups of the corresponding Hopf quiver, is clarified.     

Stacks and Torsors over Quivers

We study natural Grothendieck topologies on categories of quivers without and with relations, prove descent theorems for quiver representations, and introduce the notion of torsors over quivers.     

Embeddings of Non-semiregular Translation Quivers in Quivers of Type $\mathbb{Z}\Delta$

In this work we show that a directed translation quiver such that every path from a injective vertex to a projective vertex that has at most two hooks and in case two, they are consecutives, can be embedded in a quiver . This generalizes a result by (Li, Comm. Algebra, 28(10),4635–4645, 2000). Dedicated to Raymundo Bautista on his 60th Birthday.     

Semisimple Representations of Quivers in Characteristic p

We prove that the results of Le Bruyn and Procesi on the varieties parameterizing semisimple representations of quivers hold over an algebraically closed base field of arbitrary characteristic.