IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian category of such representations is denoted by mod-A.) Our motivation is twofold. Firstly, such moduli spaces should play an important role in organising the representation theory of wild algebras. Secondly, such moduli spaces can be identified with moduli spaces of vector bundles on special projective varieties. This identification is somewhat hidden in earner work ([6], [7]) but has become more explicit recently ([4], [12]). It can now be seen to arise from a 'tilting equivalence' between the derived category of mod-A and the derived category of coherent sheaves on the variety. It is well-established that mod-A is equivalent to the category of representations of an arrow diagram, or 'quiver', Q by linear maps satisfying certain 'admissible' relations. Thus, the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classifying orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a) of the representation space 9t{Q, a) of the quiver. Now, the moduli spaces provided by classical invariant theory ([1], [18]) are not interesting in this context. This is because the classical theory only picks out the closed GL(a)-orbits in VA{a), which correspond to semisimple /4-modules, and the quiver Q is chosen so that there is only one semisimple A -module of each dimension vector. On the other hand, we can apply Mumford's geometric invariant theory, with the trivial linearisation twisted by a character x of GL(a), which restricts our attention to an open subset of VA(a), consisting of semistable representations. Within this open set there are more closed orbits and the corresponding algebraic quotient is then a more interesting moduli space. In fact, this approach also has a classical flavour, since it involves the relative (or semi-) invariants of the GL(a) action. The main purpose of this paper is to show that the notions of semistability and stability, that arise from the geometric invariant theory, coincide with more algebraic notions, expressed in the language of mod-A Indeed, the definition is formulated for an arbitrary abelian category as follows:

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Moduli of representations of finite dimensional algebras

http://www.maths.leeds.ac.uk/~marsh/research_articles/pp18.pdf

Tilting theory and cluster combinatorics

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 modules 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 self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Mutation in triangulated categories and rigid Cohen–Macaulay modules

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit.

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Quivers with potentials and their representations II: Applications to cluster algebras

We investigate general representations of a quiver Q. There are two main questions that we shall address. The first, which was raised by Kac, is to find the dimension vectors of the summands of a general representation of dimension vector a; the second is to find the dimension vectors of the subrepresentations of a general representation of dimension vector a. These two questions are closely related. One consequence of our investigation is to find an algorithm to compute the canonical decomposition of an arbitrary dimension vector in terms of the Euler form of the quiver

General Representations of Quivers

Semi-Invariants of Quivers

On a simplical complex associated with tilting modules

Semi-invariants of quivers for arbitrary dimension vectors

Let D and E be central division algebras over k; let K be the generic splitting field of E; we show that the index of D⊗ k K is the minimum of the indices of D⊗E ○ × i as i varies. We use this to calculate the index of D under related central extensions and to construct division algebras with special properties

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Aidan Schofield

Papers

General Representations of Quivers

Semi-Invariants of Quivers

On a simplical complex associated with tilting modules

Semi-invariants of quivers for arbitrary dimension vectors

The index of a Brauer class on a Brauer-Severi variety