Tilting theory and cluster combinatorics.
type: Item
instance of: scholarly article
title: Tilting theory and cluster combinatorics.
zbMATH Open document ID: 1127.16011
author: Aslak Bakke Buan
author: Markus Reineke
author: Idun Reiten
author: Gordana Todorov
author: Bethany R. Marsh
published in: Advances in Mathematics
publication date: 2006-08-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0402054
review text: 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.
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 5047155
zbMATH Keywords: tilting modules
zbMATH Keywords: cluster algebras
zbMATH Keywords: cluster categories
zbMATH Keywords: bounded derived categories
zbMATH Keywords: module categories of finite-dimensional hereditary algebras
zbMATH Keywords: approximations
zbMATH Keywords: representation types
reviewed by: Changchang Xi
MaRDI profile type: Publication
OpenAlex ID: W1997822738
arXiv ID: math/0402054
Wikidata QID: Q56285882
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DOI: 10.1016/J.AIM.2005.06.003
description: scientific article