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Buan, Aslak Bakke; Marsh, Bethany Rose; Reineke, Markus; Reiten, Idun; Todorov, Gordana

Tilting theory and cluster combinatorics. (English) Zbl 1127.16011

Adv. Math. 204, No. 2, 572-618 (2006).
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)

Cited in 39 Reviews
Cited in 407 Documents

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 types
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\textit{A. B. Buan} et al., Adv. Math. 204, No. 2, 572--618 (2006; Zbl 1127.16011)
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