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

Published online by Cambridge University Press:  01 July 2009

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.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2009

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