We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of selfinjective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.
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.On the other hand, the connection to quantum group theory is given by the realization of quantized enveloping algebras of (Kac-Moody) Lie algebras of [Rin], [Gr] via the Hall algebra approach, which can be interpreted (see [Kap]) as a convolution algebra construction on parameter spaces of quiver representations.The aim of the present paper is to develop a synthesis of both methods. We 1 start with a particular instance of the above mentioned analogy to vector bundle theory, namely the Harder-Narasimhan recursion [HN], which was originally used for computing Betti numbers of moduli spaces.The first main result of this paper (Proposition 4.8) is a materialization of the Harder-Narasimhan recursion in the quantized enveloping algebra of a symmetric Kac-Moody Lie algebra. It leads to a canonical orthogonal system, the HN system, in such algebras (Theorem 4.9), which is recursively computable.The HN system comprises a surprising amount of information on the moduli spaces of (semi-)stable quiver representations: as the second main result, just by evaluating at a character of the quantum group, we recover in Theorem 6.7 the Betti numbers of such moduli spaces completely (modulo the standard assumption (see e.g. [Kir]) that semistability and stability coincide). The proof uses the Weil conjectures, in analogy to the original approach of Harder and Narasimhan in the vector bundle situation (see also [Kir], [Gö] for similar situations).The third main result is a resolution of the Harder-Narasimhan recursion, in the spirit of [Z], [LR] in the vector bundle case (Theorem 5.1), together with a fast algorithm for Betti number computation (Corollary 6.9). Whereas the cited works use involved explicit calculations, resp. the Langlands lemma from the theory of Eisenstein series, the present proof uses only some simple (polygonal and simplicial) combinatorics. It should be noted that our materialization of the Harder-Narasimhan recursion in a non-commutative algebra (the quantum group) is already anticipated in ([Z], p. 457), where one of the key insights for resolving the recursion is a noncommutative approach to certain polynomial expressions.At the moment, the immediate applications of the HN system to quantum group theory are largely conjectural: one can expect explicit descriptions of PBW type bases, and applications to the general structure theory of Hall algebras in the s...
Given a brane tiling, that is a bipartite graph on a torus, we can associate with it a quiver potential and a quiver potential algebra. Under certain consistency conditions on a brane tiling, we prove a formula for the Donaldson-Thomas type invariants of the moduli space of framed cyclic modules over the corresponding quiver potential algebra. We relate this formula with the counting of perfect matchings of the periodic plane tiling corresponding to the brane tiling. We prove that the same consistency conditions imply that the quiver potential algebra is a 3-Calabi-Yau algebra. We also formulate a rationality conjecture for the generating functions of the Donaldson-Thomas type invariants.
Abstract. 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
a b s t r a c tLet A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.
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