Tilting theory and cluster combinatorics |
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Authors: | Aslak Bakke Buan |
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Affiliation: | a Institutt for Matematiske Fag, Norges Teknisk-Naturvitenskapelige Universitet, N-7491 Trondheim, Norway b Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, England c Mathematisches Institut, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrae, D-48149 Münster, Germany d Department of Mathematics, Northeastern University, Boston, MA 02115, USA |
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Abstract: | We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting objects correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting. |
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Keywords: | primary 16G20 16G70 secondary 16S99 17B99 |
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