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67
Model Theory and Modules
, 2006
"... The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ..."
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Cited by 64 (20 self)
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The modeltheoretic investigation of modules has led to ideas, techniques and results which are of algebraic interest, irrespective of their modeltheoretic 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 ModR, the full subcategory of finitely presented modules will be denoted modR, the
A geometric approach to perturbation theory of matrices and matrix pencils. Part II: A stratificationenhanced staircase algorithm
 SIAM J. Matrix Anal. Appl
, 1997
"... Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a wellknown illposed 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 ..."
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Cited by 35 (8 self)
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Computing the Jordan form of a matrix or the Kronecker structure of a pencil is a wellknown illposed 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.
The combinatorics of quiver representations
 Notices Amer. Math. Soc
, 2006
"... 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 ..."
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Cited by 25 (1 self)
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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
On the quiver Grassmannian in the acyclic case
 J. Pure Appl. Algebra
"... 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 Amodule M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler chara ..."
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Cited by 21 (1 self)
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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 Amodule 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.
KAZHDAN–LUSZTIG CORRESPONDENCE FOR THE REPRESENTATION CATEGORY OF THE TRIPLET WALGEBRA IN LOGARITHMIC CFT
, 2006
"... ... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finitedimensional 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 ar ..."
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Cited by 20 (0 self)
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... is the symmetry of the (1, p) logarithmic conformal field theory model, we propose the equivalent category Cp of finitedimensional 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 quantumgroup center with the logarithmic conformal field theory center and of the universal Rmatrix with the braiding matrix.
Laurent expansions in cluster algebras via quiver representations
, 2006
"... 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 alg ..."
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Cited by 19 (2 self)
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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.
Absolutely indecomposable representations and KacMoody Lie algebras
 Invent. Math
"... 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 ..."
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Cited by 17 (1 self)
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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 KacMoody Lie algebra. In this paper we prove these conjectures for indivisible dimension vectors. 1.
Triangulated Categories and KacMoody Algebras
 Invent. math
, 2000
"... By using the RingelHall 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 2period orbit categories of the derived categories of ..."
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Cited by 15 (1 self)
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By using the RingelHall 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 2period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable KacMoody Lie algebras. 1991 AMS Subject Classification. 16G20, 17B67; Secondary 16G10, 17B37. Key Words and Phrases: triangulated category, KacMoody 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...
Auslander algebras and initial seeds for cluster algebras
 J. LONDON MATH. SOC
, 2006
"... 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 Λmodule IQ with r = Π  pairwise nonisomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of kQ to Λ. If N is ..."
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Cited by 15 (4 self)
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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 Λmodule IQ with r = Π  pairwise nonisomorphic 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.
Generic extensions and canonical bases for cyclic quivers, preprint
, 2004
"... Abstract. We use the monomial basis theory developed in [4] to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the +part U + of the quantum affine sln. This construction relies on analysis of quiver representations and the ..."
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Cited by 9 (4 self)
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Abstract. We use the monomial basis theory developed in [4] to present an elementary algebraic construction of the canonical bases for both the Ringel–Hall algebra of a cyclic quiver and the +part U + of the quantum affine sln. This construction relies on analysis of quiver representations and the introduction of a new integral PBWlike basis for the Lusztig Z[v, v −1]form of U +. Dedicated to Claus Michael Ringel on the occasion of his 60th birthday