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Cluster structures for 2-Calabi–Yau categories and unipotent groups

Part of: Homological methods Representation theory of rings and algebras Abelian categories Homological algebra Categories with structure

Published online by Cambridge University Press:  01 July 2009

A. B. Buan ,
O. Iyama ,
I. Reiten  and
J. Scott
A. B. Buan
Affiliation:
Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway (email: aslakb@math.ntnu.no)
O. Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, 464-8602 Nagoya, Japan (email: iyama@math.nagoya-u.ac.jp)
I. Reiten
Affiliation:
Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway (email: idunr@math.ntnu.no)
J. Scott
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK (email: jscott@maths.leeds.ac.uk)
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Abstract

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We investigate cluster-tilting objects (and subcategories) in triangulated 2-Calabi–Yau and related categories. In particular, we construct a new class of such categories related to preprojective algebras of non-Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi–Yau categories contains, as special cases, the cluster categories and the stable categories of preprojective algebras of Dynkin graphs. For these 2-Calabi–Yau categories, we construct cluster-tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We discuss connections with cluster algebras and subcluster algebras related to unipotent groups, in both the Dynkin and non-Dynkin cases.

Keywords

preprojective algebras2-Calabi–Yau categoriesreduced expressionsquiver mutationcluster algebrascluster categoriescluster tiltingAR-quivers

MSC classification

Secondary: 18E30: Derived categories, triangulated categories 16E99: None of the above, but in this section 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 18D99: None of the above, but in this section 18G99: None of the above, but in this section
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 1035 - 1079
Copyright
Copyright © Foundation Compositio Mathematica 2009

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