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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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Cited by 8 (7 self)
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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.
Cluster categories and selfinjective algebras: type A
, 2006
"... We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories. ..."
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Cited by 7 (2 self)
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We show that the stable module categories of certain selfinjective algebras of finite representation type having tree class An are actually ucluster categories.
TOPOLOGICAL INVARIANTS OF PIECEWISE HEREDITARY ALGEBRAS
"... Abstract. We investigate the Galois coverings of piecewise algebras and more particularly their behaviour under derived equivalences. Under a technical assumption which is satisfied if the algebra is derived equivalent to a hereditary algebra, we prove that there exists a universal Galois covering w ..."
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Cited by 2 (1 self)
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Abstract. We investigate the Galois coverings of piecewise algebras and more particularly their behaviour under derived equivalences. Under a technical assumption which is satisfied if the algebra is derived equivalent to a hereditary algebra, we prove that there exists a universal Galois covering whose group of automorphisms is free and depends only on the derived category of the algebra. As a corollary, we prove that the algebra is simply connected if and only if its first Hochschild cohomology vanishes.
ON THE STRUCTURE OF TAME GRADED BASIC HOPF ALGEBRAS II
, 903
"... Abstract. In continuation of the article [28] we classify all radically graded basic Hopf algebras of tame type over an algebraically closed field of characteristic 0. 1. ..."
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Cited by 1 (0 self)
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Abstract. In continuation of the article [28] we classify all radically graded basic Hopf algebras of tame type over an algebraically closed field of characteristic 0. 1.
STABLE CALABIYAU DIMENSION FOR FINITE TYPE SELFINJECTIVE ALGEBRAS
, 2007
"... Abstract. We show that the CalabiYau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent results of C. Amiot, and hence apply more generally ..."
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Abstract. We show that the CalabiYau dimension of the stable module category of a selfinjective algebra of finite representation type is determined by the action of the Nakayama and suspension functors on objects. Our arguments are based on recent results of C. Amiot, and hence apply more generally to triangulated categories having only finitely many indecomposable objects. Throughout, k is an algebraically closed field, and all kalgebras are finite dimensional. A selfinjective kalgebra A has a stable module category stab A. This is a triangulated category with suspension Σ given by taking the first syzygy Ω −1 A in an injective resolution. There is also a Serre functor given by S = ΩA ◦ νA where νA: = DA ⊗ − is the Nakayama functor, see [6, prop. 1.2(ii)]. By definition, the CalabiYau dimension of a triangulated category with Serre functor S is the smallest integer d ≥ 0 such that there is an equivalence of functors S ≃ Σ d, or ∞ if no d exists. CalabiYau dimensions of stable module categories of selfinjective algebras have been of considerable interest recently, see for instance [2], [3], [5], [6], [7]. In finite representation type, these dimensions occur in connection with ucluster categories of Dynkin type [9], [10], [11]. According to the definition, for determining the precise value of the CalabiYau dimension of the stable module category of a selfinjective algebra A, one needs to find the minimal d ≥ 0 such that there is an equivalence of functors νA ≃ Ω −d−1 A. In some situations it is not too difficult to show that these functors agree on objects (i.e. νA(M) ∼ = (M) for every Amodule M). On the other hand, it can be a
CLASSIFICATION OF SYMMETRIC SPECIAL BISERIAL ALGEBRAS WITH AT MOST ONE NONUNISERIAL INDECOMPOSABLE PROJECTIVE
, 2013
"... ABSTRACT. We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one nonuniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence ..."
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ABSTRACT. We consider a natural generalisation of symmetric Nakayama algebras, namely, symmetric special biserial algebras with at most one nonuniserial indecomposable projective module. We describe the basic algebras explicitly by quiver and relations, then classify them up to derived equivalence and up to stable equivalence of Morita type. This includes the algebras of [5], where they study the weakly symmetric algebras of Euclidean type, as well as some algebras of dihedral type.