Elements of the Representation Theory of Associative Algebras

Assem, Ibrahim; Skowroński, Andrzej; Simson, Daniel

doi:10.1017/cbo9780511614309

Cited by 1,268 publications

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“…Recall that a triangular algebra A is said to be p q -Calabi-Yau for integers q ≥ 1 and p ∈ Z if S q = [p] in the derived category Der(A), where S = τ • [1]. We show that algebras satisfying the fractional Calabi-Yau property have periodic Coxeter transformation and are, therefore, of cyclotomic type, see Sections 2 and 5.…”

Section: Introductionmentioning

confidence: 92%

Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Peña

2013

Algebr Represent Theor

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Abstract. Let A be a finite dimensional algebra over an algebraically closed field k. Assume A is basic connected with n pairwise non-isomorphic simple modules. We consider the Coxeter transformation φ A as the automorphism of the Grothendieck group K 0 (A) induced by the Auslander-Reiten translation τ in the derived category Der(modA) of the module category modA of finite dimensional left A-modules. We say that A is an algebra of cyclotomic type if the characteristic polynomial χ A of φ A is a product of cyclotomic polynomials. There are many examples of algebras of cyclotomic type in the representaton theory literature: hereditary algebras of Dynkin and extended Dynkin types, canonical algebras, some supercanonical and extended canonical algebras. Among other results, we show that: (a) algebras satisfying the fractional Calabi-Yau property have periodic Coxeter transformation and are, therefore, of cyclotomic type, and (b) algebras whose homological form h A is non-negative are of cyclotomic type. For an algebra A of cyclotomic type we describe the shape of the Auslander-Reiten components of Der(modA).

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Section: Introductionmentioning

confidence: 92%

Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Peña

2013

Algebr Represent Theor

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“…We refer to the books [64] [33] [2] and [1] for a wealth of information on the representation theory of quivers and finite-dimensional algebras. Here, we will only need very basic notions.…”

Section: Categorificationmentioning

confidence: 99%

Categorification of Acyclic Cluster Algebras: An Introduction

Keller

2010

Higher Structures in Geometry and Physics

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Summary. This is a concise introduction to Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category. MSC 2010 classification: 18E30, 16S99

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“…For background on representation theory of finite dimensional modules, we refer to [5,6,32], of infinite dimensional modules to [15,26,31], and on wild hereditary algebras to [23].…”

Section: Theorem a Let H Be A Connected Hereditary Artin Algebra Of mentioning

confidence: 99%

Constructing tilting modules

Kerner

Trlifaj

2007

Trans. Amer. Math. Soc.

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Abstract. We investigate the structure of (infinite dimensional) tilting modules over hereditary artin algebras. For connected algebras of infinite representation type with Grothendieck group of rank n, we prove that for each 0 ≤ i < n − 1, there is an infinite dimensional tilting module T i with exactly i pairwise non-isomorphic indecomposable finite dimensional direct summands. We also show that any stone is a direct summand in a tilting module. In the final section, we give explicit constructions of infinite dimensional tilting modules over iterated one-point extensions.The study of finite dimensional tilting modules of projective dimension at most one over finite dimensional algebras was initiated by Brenner and Butler [11] and continued by Happel and Ringel [17]. Since then, many variations of this concept have been introduced and used successfully, for example: Tilting modules of higher projective dimension, tilting modules over rings, tilting complexes in derived categories, tilting objects in hereditary categories or in cluster categories. In this paper, we use the term tilting module as follows: Let R be a ring and T be a right R-module. Then T is a tilting module provided that (T1) p.dimT ≤ 1, (T2) Ext 1 R (T, T (I) ) = 0 for any set I, and (T3) there is a short exact sequence 0 → R → T 0 → T 1 → 0 where T 0 and T 1 are direct summands in a direct sum of (possibly infinitely many) copies of T . Equivalently, T is tilting if and only if Gen(T ) = {T } ⊥ , [13].Here, Gen(T ) denotes the class of all homomorphic images of direct sums of copies of T , and, for a class of modules C,If T is a tilting module, then {T } ⊥ is a torsion class in Mod-R, the tilting class generated by T . If T is another tilting module, then T is said to be equivalent toThough our definition of a tilting module allows infinitely generated modules, there is an implicit finiteness property connected with tilting, recently proved by Bazzoni and Herbera in [9, Theorem 2.4]. Namely, any tilting module T is of finite type, that is, there exists a set S consisting of finitely presented modules of projective dimension at most 1 such that S ⊥ = {T } ⊥ .

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Elements of the Representation Theory of Associative Algebras

Cited by 1,268 publications

References 0 publications

Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Algebras Whose Coxeter Polynomials are Products of Cyclotomic Polynomials

Categorification of Acyclic Cluster Algebras: An Introduction

Constructing tilting modules

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