Tilting theory and cluster combinatorics.(English)Zbl 1127.16011
From the authors’ summary: We introduce a new category \(\mathcal C\), which we call the cluster category, obtained as a quotient of the bounded derived category \(\mathcal D\) of the module category of a finite-dimensional hereditary algebra \(H\) over a field. We show that, in the simply laced Dynkin case, \(\mathcal C\) can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. Using approximation theory, we investigate the tilting theory of \(\mathcal C\), 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.
Reviewer: Xi Changchang (Beijing)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
| 16D90 | Module categories in associative algebras |
Keywords:
tilting modules;cluster algebras;cluster categories;bounded derived categories;module categories of finite-dimensional hereditary algebras;approximations;representation typesCite
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