Markus Reineke

Generalized associahedra via quiver representations

We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.

The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

Abstract.Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in ‘linear algebra type’ situations.

Poisson automorphisms and quiver moduli

A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulas for Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman.

Moduli of representations of quivers

This is a survey article about properties of Cohen-Macaulay modules over surface singularities. We discuss properties of the Macaulayfication functor, reflexive modules over simple, quotient and minimally elliptic singularities, geometric and algebraic McKay Corre- spondence. Finally, we describe matrix factorizations corresponding to indecomposable Cohen- Macaulay modules over the non-isolated singularities A1 and D1.For a finite dimensional algebra A of finite global dimension the bounded derived category of finite dimensional A-modules admits Auslander- Reiten triangles such that the Auslander-Reiten translation τ is an equivalence. On the level of the Grothendieck group τ induces the Coxeter transformation �A. More generally this extends to a homologically finite triangulated category T admitting Serre duality. In both cases the Coxeter polynomial, that is, the characteristic polynomial of the Coxeter transformation yields an important homological invariant of A or T. Spectral analysis is the study of this interplay, it often reveals unexpected links between apparently different subjects. This paper gives a summary on spectral techniques and studies the links to singularity theory. In particular, it offers a contribution to the categorifica- tion of the Milnor lattice through triangulated categories which are naturally attached to a weighted projective line.We review the definition of a Calabi-Yau triangulated category and survey examples coming from the representation theory of quivers and finite-dimensional algebras. Our main motivation comes from the links between quiver representations and Fomin-Zelevinsky’s cluster algebras. Mathematics Subject Classification (2000). Primary 18E30; Secondary 16D90, 18G10.The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.An introduction to moduli spaces of representations of quivers is given, and results on their global geometric properties are surveyed. In particular, the geometric approach to the problem of classification of quiver representations is motivated, and the construction of moduli spaces is reviewed. Topological, arithmetic and algebraic methods for the study of moduli spaces are discussed.We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce

Smooth models of quiver moduli

n

Counting rational points of quiver moduli

-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.

Quivers, Desingularizations and Canonical Bases

For any moduli space of stable representations of quivers, certain smooth varieties, compactifying projective space fibrations over the moduli space, are constructed. The boundary of this compactification is analyzed. Explicit formulas for the Betti numbers of the smooth models are derived. In the case of moduli of simple representations, explicit cell decompositions of the smooth models are constructed.

Quiver Grassmannians and degenerate flag varieties

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra of a quiver.

Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

A class of desingularizations for orbit closures of representations of Dynkin quivers is constructed, which can be viewed as a graded analogue of the Springer resolution. A stratification of the singular fibres is introduced; its geometry and combinatorics are studied. Via the Hall algebra approach, these constructions relate to bases of quantized enveloping algebras. Using Ginzburg’s theory of convolution algebras, the base change coefficients of Lusztig’s canonical basis are expressed as decomposition numbers of certain convolution algebras.

Monomials in canonical bases of quantum groups and quadratic forms

Quiver Grassmannians are varieties parametrizing subrepresentations of a quiver representation. It is observed that certain quiver Grassmannians for type A quivers are isomorphic to the degenerate flag varieties investigated earlier by the second named author. This leads to the consideration of a class of Grassmannians of subrepresentations of the direct sum of a projective and an injective representation of a Dynkin quiver. It is proven that these are (typically singular) irreducible normal local complete intersection varieties, which admit a group action with finitely many orbits, and a cellular decomposition. For type A quivers explicit formulas for the Euler characteristic (the median Genocchi numbers) and the Poincare polynomials are derived.