The model-theoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their model-theoretic significance. It is these aspects that I will discuss in this article, although I will make some comments on the model theory of modules per se. Our default is that the term “module ” will mean (unital) right module over a ring (associative with 1) R. The category of such modules is denoted Mod-R, the full subcategory of finitely presented modules will be denoted mod-R, the

Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a well-known ill-posed problem. We propose that knowledge of the closure relations, i.e., the stratification, of the orbits and bundles of the various forms may be applied in the staircase algorithm. Here we discuss and complete the mathematical theory of these relationships and show how they may be applied to the staircase algorithm. This paper is a continuation of our Part I paper on versal deformations, but may also be read independently.

... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finite-dimensional representations of the restricted quantum group Uqsℓ(2) at q = e iπ p. We fully describe the category Cp by classifying all indecomposable representations. These are exhausted by projective modules and three series of representations that are essentially described by indecomposable representations of the Kronecker quiver. The equivalence of the W(p)- and Uqsℓ(2)-representation categories is conjectured for all p �2 and proved for p = 2, the implications including the identifications of the quantum-group center with the logarithmic conformal field theory center and of the universal R-matrix with the braiding matrix.

A quiver is just a directed graph. 1 Formally, a quiver is a pair Q=(Q0,Q1) where Q0 is a finite set of vertices and Q1 is a finite set of arrows between them. If a∈Q1 is an arrow, then ta and ha

Dedicated to Idun Reiten on the occasion of her sixtieth birthday. Abstract. A conjecture of Kac states that the polynomial counting the number of absolutely indecomposable representations of a quiver over a finite field with given dimension vector has positive coefficients and furthermore that its constant term is equal to the multiplicity of the corresponding root in the associated Kac-Moody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors. 1.

Abstract. Let 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. Let M be a finite dimensional space on a field k. The Grassmannian Gre(M,k) of M is the set of subspaces of dimension e. It is well known that Gre(M,k) is an algebraic variety with nice properties. For instance, the linear group GLe(M,k) acts transitively on Gre(M,k) with parabolic stabilizer, hence the variety Gre(M,k) is smooth and projective.

To Alexander Alexandrovich Kirillov on the occasion of his seventieth birthday Abstract. We study Laurent expansions of cluster variables in a cluster algebra of rank 2 associated to a generalized Kronecker quiver. In the case of the ordinary Kronecker quiver, we obtain explicit expressions for Laurent expansions of the elements of the canonical basis for the corresponding cluster algebra. 1.

Abstract. Let Q be a Dynkin quiver and Π the corresponding set of positive roots. For the preprojective algebra Λ associated to Q we produce a rigid Λ-module IQ with r = |Π | pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of kQ to Λ. If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type |Q|, then the coordinate ring C[N] is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of IQ coincide with certain generalized minors which form an initial cluster for C[N], and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of EndΛ(IQ). Finally, we exploit the fact that the categories of injective modules over Λ and over its covering ˜ Λ are triangulated in order to show several interesting identities in the respective stable module categories.

By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T 2 = 1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable Kac-Moody Lie algebras. 1991 AMS Subject Classification. 16G20, 17B67; Secondary 16G10, 17B37. Key Words and Phrases: triangulated category, Kac-Moody algebra, derived category, hereditary algebra. Contents 1 Introduction 1.1 A Cartan datum is a pair (I; (; )) consisting of a finite set I and a symmetric bilinear form (\Gamma; \Gamma) on the free abelian group Z[I] with values in Z such that the following conditions are satisfied. (a) (i; i) 2 f2; 4; 6; \Delta \Delta \Deltag for any i 2 I: (b) 2(i; j)=(i; i) 2 f0; \Gamma1; \Gamma2; \Delta \Delta \Deltag for any i 6= j in I: Then C = (a ij ) i;j...

By introducing Frobenius morphisms F on algebras A and their modules over the algebraic closure F q of the finite field F q of q elements, we establish a relation between the representation theory of A over F q and that of the F -fixed point algebra A over F q . More precisely, we prove that the category mod-A of finite dimensional A -modules is equivalent to the subcategory of finite dimensional F -stable A-modules, and, when A is finite dimensional, we establish a bijection between the isoclasses of indecomposable A -modules and the F -orbits of the isoclasses of indecomposable A-modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over F q can be interpreted as F -stable representations of a corresponding quiver over F q . We further prove that every finite dimensional hereditary algebra over F q is Morita equivalent to some A , where A is the path algebra of a quiver Q over F q and F is induced from a certain automorphism of Q. A close relation between the Auslander-Reiten theories for A and A is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of A is obtained by "folding" the Auslander-Reiten quiver of A. Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver with a given dimension vector and to establish part of Kac's theorem for all finite dimensional hereditary algebras over a finite field. CONTENTS 1.