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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 ..."
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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.
Applications of BGPreflection functors: isomorphisms of cluster algebras
"... Abstract. Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to t ..."
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Abstract. Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1, · · · , un) of rational functions of n independent indeterminates u1, · · · , un. It is an isomorphism between two cluster algebras associated to the matrix A (see section 4 for precise meaning). When A is of finite type, these isomorphisms behave nicely, they are compatible with the BGPreflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the ”truncated simple reflections” defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by DlabRingel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types. Key words. Coxeter automorphisms of cluster algebras, BGPreflection functors, cluster variables. Mathematics Subject Classification. 16G20, 16G70, 19S99, 17B20. 1.
FINITE AND BOUNDED AUSLANDERREITEN COMPONENTS IN THE DERIVED CATEGORY
, 2009
"... We analyze AuslanderReiten components for the bounded derived category of a finitedimensional algebra. We classify derived categories whose AuslanderReiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Ausland ..."
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We analyze AuslanderReiten components for the bounded derived category of a finitedimensional algebra. We classify derived categories whose AuslanderReiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their AuslanderReiten quiver is determined. We also determine components that contain shift periodic complexes.
AUSLANDERREITEN COMPONENTS IN THE BOUNDED DERIVED CATEGORY
, 2009
"... Derived Categories of finite dimensional algebras whose AuslanderReiten quiver has a finite component, a component with Dynkin tree class or a bounded component are classified. Their AuslanderReiten quiver is determined. We use these results to show that certain algebras are piecewise hereditary. ..."
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Derived Categories of finite dimensional algebras whose AuslanderReiten quiver has a finite component, a component with Dynkin tree class or a bounded component are classified. Their AuslanderReiten quiver is determined. We use these results to show that certain algebras are piecewise hereditary. A necessary condition for components of Euclidean tree class is deduced and components that contain shift periodic complexes are determined.
A NEW TRIANGULATED CATEGORY FOR RATIONAL SURFACE SINGULARITIES
, 2009
"... In this short paper we introduce a new triangulated category for rational surface singularities which in the nonGorenstein case acts as a substitute ..."
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In this short paper we introduce a new triangulated category for rational surface singularities which in the nonGorenstein case acts as a substitute
MUTATING LOOPS AND 2CYCLES IN 2CY TRIANGULATED CATEGORIES
"... Abstract. We derive a method for mutating quivers of 2CY tilted algebras that have loops and 2cycles, under certain specific conditions. Further, we give the classification of the 2CY tilted algebras coming from standard algebraic 2CY triangulated categories with a finite number of indecomposabl ..."
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Abstract. We derive a method for mutating quivers of 2CY tilted algebras that have loops and 2cycles, under certain specific conditions. Further, we give the classification of the 2CY tilted algebras coming from standard algebraic 2CY triangulated categories with a finite number of indecomposables. These algebras satisfy the setup for our method of mutation.
Journal of Pure and Applied Algebra 209 (2007) 497–506 www.elsevier.com/locate/jpaa BGPreflection functors and cluster combinatoricsI
, 2006
"... We define Bernstein–Gelfand–Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections ” on the set of almost positive roots Φ≥−1 associated with a finite dimensional s ..."
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We define Bernstein–Gelfand–Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the “truncated simple reflections ” on the set of almost positive roots Φ≥−1 associated with a finite dimensional semisimple Lie algebra. Combining this with the tilting theory in cluster categories developed in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054], we give a unified interpretation via quiver representations for the generalized associahedra associated with the root systems of all Dynkin types (simply laced or nonsimply laced). This confirms the Conjecture 9.1 in [A. Buan, R. Marsh, M. Reineke, I. Reiten, G. Todorov, Tilting theory and cluster combinatorics, Adv. Math. (in press). math.RT/0402054] for