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).     

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.     

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.     

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.     

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.     

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.     

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.     

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].     

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.     

Topology of character varieties and representations of quivers

In Hausel et al. (2008) [10] we presented a conjecture generalizing the Cauchy formula for Macdonald polynomial. This conjecture encodes the mixed Hodge polynomials of the character varieties of representations of the fundamental group of a punctured Riemann surface of genus g. We proved several results which support this conjecture. Here we announce new results which are consequences of those in Hausel et al. (2008) [10].     

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.     

The moment mapping for unitary representations

For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping.     

Morse theory indomitable

Dedicated to René Thom.