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COMBINATORIAL DERIVED INVARIANTS FOR GENTLE ALGEBRAS
, 2006
"... Abstract. We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stab ..."
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Abstract. We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable AuslanderReiten quiver of its repetitive algebra. As a byproduct we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.
Derived classification of gentle algebras with two cycles
"... Abstract. We classify gentle algebras defined by quivers with two cycles under derived equivalence in a non degenerate case, by using some combinatorial invariants constructed from the quiver with relations defining these algebras. We also present a list of normal forms; any such algebra is derived ..."
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Abstract. We classify gentle algebras defined by quivers with two cycles under derived equivalence in a non degenerate case, by using some combinatorial invariants constructed from the quiver with relations defining these algebras. We also present a list of normal forms; any such algebra is derived equivalent to one of the algebras in the list. The article includes an Appendix presenting a slightly modified and extended version of a technical result in the unpublished manuscript [HSZ01] by Holm, Schröer and Zimmermann, describing some essential elementary transformations over the quiver with relations defining the algebra.
DISCRETE DERIVED CATEGORIES I HOMOMORPHISMS, AUTOEQUIVALENCES AND TSTRUCTURES
"... Abstract. Discrete derived categories were introduced by Vossieck [35] and classified by Bobiński, Geiß, Skrowoński [8]. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded tstructures. Contents ..."
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Abstract. Discrete derived categories were introduced by Vossieck [35] and classified by Bobiński, Geiß, Skrowoński [8]. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded tstructures. Contents
On derived equivalence classification of gentle twocycle algebras
 Colloq. Math
"... Abstract. We classify, up to derived (equivalently, tiltingcotilting) equivalence all nondegenerate gentle twocycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case. Introduction and the main result Throughout the paper k denotes a fixed algebraical ..."
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Abstract. We classify, up to derived (equivalently, tiltingcotilting) equivalence all nondegenerate gentle twocycle algebras. We also give a partial classification and formulate a conjecture in the degenerate case. Introduction and the main result Throughout the paper k denotes a fixed algebraically closed field. By an algebra we mean a finite dimensional basic connected kalgebra and by a module a finite dimensional left module. By Z, N, and N+, we denote the sets of integers, nonnegative integers, and positive integers,
WHICH CANONICAL ALGEBRAS ARE DERIVED EQUIVALENT TO INCIDENCE ALGEBRAS OF POSETS?
, 708
"... Abstract. We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras whose number of weights is either 2 or 3. This note concerns the characterization of the canonica ..."
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Abstract. We give a full description of all the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite posets. These are the canonical algebras whose number of weights is either 2 or 3. This note concerns the characterization of the canonical algebras over an algebraically closed field that are derived equivalent to incidence algebras of finite partially ordered sets (posets), expressed in the following theorem. Theorem. Let Λ be a canonical algebra of type (p,λ) over an algebraically closed field. Then Λ is derived equivalent to an incidence algebra of a poset if and only if the number of weights of p is either 2 or 3. This theorem can be interpreted both geometrically and algebraically. From a geometric viewpoint, by considering modules over incidence algebras as sheaves over finite spaces [8] and using the derived equivalence between the categories of modules over a canonical algebra and coherent sheaves over a weighted projective line [3], we are able to obtain explicit derived