Results 1  10
of
10
Cluster structures for 2CalabiYau categories and unipotent groups
"... Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This c ..."
Abstract

Cited by 32 (6 self)
 Add to MetaCart
Abstract. We investigate cluster tilting objects (and subcategories) in triangulated 2CalabiYau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of nonDynkin quivers associated with elements in the Coxeter group. This class of 2CalabiYau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2CalabiYau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
"... Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). I ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
Abstract. This is an introduction to some aspects of FominZelevinsky’s cluster algebras and their links with the representation theory of quivers and with CalabiYau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences. Contents
Derived equivalences from mutations of quivers with potential
 ADVANCES IN MATHEMATICS 226 (2011) 2118–2168
, 2011
"... ..."
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
Abstract

Cited by 8 (7 self)
 Add to MetaCart
Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
The anticyclic operad of moulds
 International Mathematics Research Notices
, 2007
"... A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are introduced and studied. Under some restrictions, a forgetful map ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
A new anticyclic operad Mould is introduced, on spaces of functions in several variables. It is proved that the Dendriform operad is an anticyclic suboperad of this operad. Many operations on the free Mould algebra on one generator are introduced and studied. Under some restrictions, a forgetful map from moulds to formal vector fields is then defined. A connection to the theory of tilting modules for quivers of type A is also described. 0
AuslanderReiten theory revisited
 In Trends in representation theory of algebras and related topics, EMS Ser. Congr. Rep
, 2008
"... Abstract. We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained. ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. We recall several results in AuslanderReiten theory for finitedimensional algebras over fields and orders over complete local rings. Then we introduce ncluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.
UNIVERSAL DERIVED EQUIVALENCES OF POSETS OF TILTING MODULES
, 708
"... Abstract. We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flipflop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers wit ..."
Abstract
 Add to MetaCart
Abstract. We show that for two quivers without oriented cycles related by a BGP reflection, the posets of their tilting modules are related by a simple combinatorial construction, which we call flipflop. We deduce that the posets of tilting modules of derived equivalent path algebras of quivers without oriented cycles are universally derived equivalent. 1.
RECONSTRUCTION OF PATH ALGEBRAS FROM THEIR POSETS OF TILTING MODULES
"... Abstract. Let Λ = k − → ∆ be the path algebra of a finite quiver without oriented cycles. The set of isomorphism classes of multiplicity free tilting modules is in a natural way a partially ordered set. We will show here that TΛ uniquely determines − → ∆if − → ∆ has no multiple arrows and no isola ..."
Abstract
 Add to MetaCart
Abstract. Let Λ = k − → ∆ be the path algebra of a finite quiver without oriented cycles. The set of isomorphism classes of multiplicity free tilting modules is in a natural way a partially ordered set. We will show here that TΛ uniquely determines − → ∆if − → ∆ has no multiple arrows and no isolated vertices. Let Λ be a basic, connected finite dimensional algebra over an algebraically closed field k and let mod Λ be the category of finitely generated left Λmodules. For a module M ∈ mod Λ we denote by pd ΛM the projective dimension of M and by gl.dim Λ the global dimension of Λ. A module T ∈ mod Λ is called a tilting module if the following three conditions are satisfied: (i) pd ΛT<∞, (ii) Ext i Λ(T,T)=0for all i>0 and (iii) there exists an exact sequence 0 → ΛΛ → T 0 →···→T r → 0with T i ∈ add T for all 0 ≤ i ≤ r, where add T is the full subcategory of mod Λ whose objects are direct sums of direct summands of T. We will say that a tilting module is basic or multiplicity free if in a direct sum decomposition of T the indecomposable direct summands of T occur with multiplicity one. Unless stated otherwise all tilting modules considered here will be assumed to be basic.
Contents
, 2013
"... Abstract. Let T be a tilting object in a triangulated category which is equivalent to the bounded derived category of a finitedimensional hereditary algebra. The text investigages the strong global dimension, in the sense of Ringel, of the opposite algebra A of the endomorphism algebra of T. This i ..."
Abstract
 Add to MetaCart
Abstract. Let T be a tilting object in a triangulated category which is equivalent to the bounded derived category of a finitedimensional hereditary algebra. The text investigages the strong global dimension, in the sense of Ringel, of the opposite algebra A of the endomorphism algebra of T. This invariant is expressed in terms of the lengths of the sequences T0,...,Tℓ of tilting objects such that Tℓ = T, each term arises from the preceding one by a tilting mutation, and the opposite of the endomorphism algebra of T0 is a tilted algebra. It is also expressed in terms on the hereditary abelian subcategories of the triangulated category.