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12
Algebras from surfaces without punctures
"... Abstract. We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface alge ..."
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Abstract. We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial clustertilting objects in generalized cluster categories, we compute the invariant of AvellaAlaminos and Geiss for surface algebras and we provide a
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 The article includes an Appendix presenting a slightly modified and extended version of a technical result in the unpublished manuscript [HSZ01] by Holm, Schröer and Zimmermann, describing some essential elementary transformations over the quiver with relations defining the algebra.
Weighted locally gentle quivers and Cartan matrices, preprint 2005
"... Abstract. We study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of nitedimensional algebras, to a wider class of potentially innitedimensional algebras. Weights ..."
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Abstract. We study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of nitedimensional algebras, to a wider class of potentially innitedimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For the natural grading by path lengths, any locally gentle algebra is a Koszul algebra. Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. This determinant is invariant under graded derived equivalences of the corresponding algebras. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver, more precisely by the number and the weight of certain oriented cycles. This leads to combinatorial invariants of the graded derived categories of graded locally gentle algebras. By specializing to certain parameters we reobtain and extend in this way results from [4] on Cartan determinants of gentle algebras. 1.
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,
2 DERIVED EQUIVALENCE OF SURFACE ALGEBRAS IN GENUS 0 VIA GRADED EQUIVALENCE
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THE AGINVARIANT FOR (m+ 2)ANGULATIONS
"... Abstract. In this paper, we study gentle algebras that come from (m + 2)angulations of unpunctured Riemann surfaces with boundary and marked points. We focus on calculating a derived invariant introduced by AvellaAlaminos and Geiss, generalizing previous work done when m = 1. In particular, we pr ..."
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Abstract. In this paper, we study gentle algebras that come from (m + 2)angulations of unpunctured Riemann surfaces with boundary and marked points. We focus on calculating a derived invariant introduced by AvellaAlaminos and Geiss, generalizing previous work done when m = 1. In particular, we provide a method for calculating this invariant based on the the configuration of the arcs in the (m + 2)angulation, the marked points, and the boundary components. 1.