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.     

Regular Representations of the Generalized Heisenberg Algebra

We investigate general properties of regular irreducible representations of the generalized Heisenberg algebra in an arbitrary nondegenerate weakly closed space with an indefinite metric.     

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 ?.     

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.     

Geometry of the Moment Map for Representations of Quivers

We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden–Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero.     

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.     

Positivity for Generalized Cluster Variables of Affine Quivers

It has been proved in Lee and Schiffler, Ann. of Math. 182(1) 73–125 2015 that cluster variables of all skew-symmetric cluster algebras are positive. i.e., every cluster variable as a Laurent polynomial in the cluster variables of any fixed cluster has positive coefficients. We prove that every regular generalized cluster variable of an affine quiver is positive. As a corollary, we obtain that generalized cluster variables of affine quivers are positive and we also construct various positive bases. This generalizes the results in Dupont, J. Algebra Appl. 11(4) 19 2012 and Ding et al. Algebr. Represent. Theory 16(2) 491–525 2013.     

Decomposition of Marsden–Weinstein Reductions for Representations of Quivers

We decompose the Marsden–Weinstein reductions for the moment map associated to representations of a quiver. The decomposition involves symmetric products of deformations of Kleinian singularities, as well as other terms. As a corollary we deduce that the Marsden–Weinstein reductions are irreducible varieties.     

Integral Bases of Cluster Algebras and Representations of Tame Quivers

In Caldero and Keller (Invent Math 172:169–211, 2008) and Sherman and Zelevinsky (Mosc Math J 4(4):947–974, 2004), the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$ , respectively. In this paper, we will deduce ?-bases for cluster algebras of affine types.     

广义逆矩阵的张量积

本文证明了广义逆矩阵张量积的一些性质,介绍了它在解线性方程组方面的应用,并得到了矩阵张量积的奇异值的一些性质     

Existence Theorems for Generalized Moment Representations

We establish conditions for the existence of generalized moment representations introduced by Dzyadyk in 1981.     

Application of Full Quivers of Representations of Algebras,to Polynomial Identities

In [7 Belov , A. , Rowen , L. H. , Vishne , U. Full quivers of representations of algebras. To appear in Trans. Amer. Math. Soc.  [Google Scholar]] we introduced the notion of full quivers of representations of algebras, which are more explicit than quivers of algebras, and better suited for algebras over finite fields. Here, we consider full quivers as a combinatorial tool in order to describe PI-varieties of algebras. We apply the theory to clarify the proofs of diverse topics in the literature: Determining which relatively free algebras are weakly Noetherian, determining when relatively free algebras are finitely presented, presenting a quick proof for the rationality of the Hilbert series of a relatively free PI-algebra, and explaining counterexamples to Specht's conjecture for varieties of Lie algebras.     

Representations of Generalized Almost-Jordan Algebras

This paper deals with the variety of commutative algebras satisfying the identity β{(yx ²)x ? ((yx)x)x} + γ{yx ³ ? ((yx)x)x} = 0, where β, γ are scalars. These algebras appeared as one of the four families of degree four identities in Carini, Hentzel, and Piacentini-Cattaneo [6 Carini , L. , Hentzel , I. R. , Piacentini-Cattaneo , J. M. ( 1988 ). Degree four identities not implies by commutativity . Comm. in Algebra 16 ( 2 ): 339 – 357 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. We give a characterization of representations and irreducible modules on these algebras. Our results require that the characteristic of the ground field is different from 2, 3.     

The Norm of a Relation,Separating Functions,and Representations of Marked Quivers

We consider numerical functions that characterize Dynkin schemes, Coxeter graphs, and tame marked quivers.     

Rational Invariants of the Generalized Classical Groups

In this paper,we give transcendence bases of the rational invariants fields of the generalized classical groups and their subgroups B,N and T,and we also compute the orders of them.Furthermore,we give explicit generators for the rational invariants fields of the Borel subgroup and the Neron-Severi subgroup of the general linear group.     

On Symmetric Invariants of Level Surfaces Near Regular Points

We consider the symmetric invariants of the level surfaces ofa smooth function away from its critical points, and prove forthem some formulae in divergence from. We then apply these formulaeto obtain an isoperimetric inequality for the surface area oflevel surfaces of p-capacity potentials.     

Trace and Density Results on Regular Trees

Potential Analysis - The boundary of a regular tree can be viewed as a Cantor-type set. We equip our tree with a weighted distance and a weighted measure via the Euclidean arc-length and consider...