Locally semi-simple representations of quivers

We suggest a geometrical approach to the semi-invariants of quivers based on Luna's slice theorem and the Luna-Richardson theorem. The locally semi-simple representations are defined in this spirit but turn out to be connected with stable representations in the sense of GIT, Schofield's perpendicular categories, and Ringel's regular representations. As an application of this method we obtain an independent short proof of a theorem of Skowronski and Weyman about semi-invariants of the tame quivers.     

Asymptotic degeneration of representations of quivers

We define asymptotic degeneration of nilpotent representations of an arbitrary finite quiver, using large tensor powers and small direct sums, and characterize this notion by a simple and effective criterion.     

Semi-invariants of mixed representations of quivers

A notion of a mixed representation of a quiver can be derived from ordinary quiver representation by considering the dual action of groups on "vertex" vector spaces together with their usual action. A generating system for the algebra of semi-invariants of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers of a special form and using a function DP on three matrices, which is a mixture of the determinant and two pfaffians.     

Rings of invariants for representations of quivers

In this Note we compute the generators of the ring of invariants for quiver factorization problems, generalizing results of Le Bruyn and Procesi. In particular, we find a necessary and sufficient combinatorial criterion for the projectivity of the associated invariant quotients. Further, we show that the non-projective quotients admit open immersions into projective varieties, which still arise from suitable quiver factorization problems. To cite this article: M. Halic, M.S. Stupariu, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On spherical representations of quivers and generalized complexes

For any quiverQ we consider spherical representationsV ofQ such that the isomorphism class ofV is a spherical variety. We suggest an approach for classifying such representations for anyQ and obtain a classification forQ being an equioriented Dynkin diagramA _n. In particular, all complexes are spherical representations. We introduce a category of representations that we call generalized complexes and classify spherical generalized complexes. For the quivers that we call crumbly we prove that any spherical generalized complex has a polynomial algebra of covariants on the closure of its isomorphism class.Partially supported by INTAS-OPEN grant 97-1570 and RFFI grant 98-01-00598.     

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.     

Normality of Marsden-Weinstein reductions for representations of quivers

 We prove that the Marsden-Weinstein reductions for the moment map associated to representations of a quiver are normal varieties. We give an application to conjugacy classes of matrices. Received: 10 August 2001 / Published online: 16 October 2002 Mathematics Subject Classification (2000): 16G20, 53D20, 14B05.     

t-structures in the derived category of representations of quivers

Given a finite quiver without oriented cycles, we describe a family of algebras whose module category has the same derived category as that of the quiver algebra. This is done in the more general setting oft-structures in triangulated categories. A completeness result is shown for Dynkin quivers, thus reproving a result of Happel [H].     

G-exceptional vector bundles on P2 and representations of quivers

It is known that the category of homogeneous bundles on ${\mathbb{P}}^2$ is equivalent to the category of representations of a quiver with relation. In this paper we make use of this equivalence to describe a family of G-exceptional bundles on ${\mathbb{P}}^2$ and to prove that they are stable. We also study the G-exceptionality of Fibonacci bundles on ${\mathbb{P}}^2$.     

Codimension two singularities for representations of extended Dynkin quivers

Let M and N be two representations of an extended Dynkin quiver such that the orbit of N is contained in the orbit closure and has codimension two. We show that the pointed variety is smoothly equivalent to a simple surface singularity of type A _n , or to the cone over a rational normal curve.     

一类D型箭图的表示的Frobenius-Perron维数

本文通过研究一类D型箭图的不可分解表示的砖块(brick)集,给出这些不可分解表示的Frobenius-Perron维数.     

Matrix factorizations and representations of quivers II: Type ADE case

We study a triangulated category of graded matrix factorizations for a polynomial of type ADE. We show that it is equivalent to the derived category of finitely generated modules over the path algebra of the corresponding Dynkin quiver. Also, we discuss a special stability condition for the triangulated category in the sense of T. Bridgeland, which is naturally defined by the grading.     

Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder–Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder–Narasimhan–Jordan–Hölder filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.     

Orthoscalar representations of quivers in the category of Hilbert spaces

As is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers correspond to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article “Locally scalar representations of graphs in the category of Hilberts spaces” (Func. Anal. Apps., 2005), the authors showed a way for carrying over these results to Hilbert spaces, constructed Coxeter functors, and proved an analog of the Gabriel theorem for locally scalar representations (up to unitary equivalence). The category of locally scalar representations of a quiver can be regarded as a subcategory in the category of all representations (over the field ℂ). In the present paper, we study the relationship between the indecomposability of locally scalar representations in the subcategory and in the category of all representations (it is proved that for a class of quivers wide enough indecomposability in the subcategory implies indecomposability in the category). For a quiver corresponding to the extended Dynkin graph , locally scalar representations that cannot be obtained from the simplest ones by Coxeter functors (regular representations) are classified. Bibliography: 21 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 180–201.     

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.     

Singular locally scalar representations of quivers in Hilbert spaces and separating functions

We consider locally scalar representations of extended Dynkin graphs in Hilbert spaces. The relation between these representations and the function (n) = 1 + (n – 1)/(n + 1) is established. We construct a family of separating functions that generalize the function and play a similar role in a broader class of graphs.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 796–809, June, 2004.