Compositio Mathematica

Research Article

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

A. B. Buana1, O. Iyamaa2, I. Reitena3 and J. Scotta4

a1 Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway (email: aslakb@math.ntnu.no)

a2 Graduate School of Mathematics, Nagoya University, Chikusa-ku, 464-8602 Nagoya, Japan (email: iyama@math.nagoya-u.ac.jp)

a3 Institutt for matematiske fag, Norges teknisk-naturvitenskapelige universitet, N-7491 Trondheim, Norway (email: idunr@math.ntnu.no)

a4 Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, UK (email: jscott@maths.leeds.ac.uk)

Abstract

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.

(Received November 06 2007)

(Accepted October 21 2008)

(Online publication May 13 2009)

2000 Mathematics Subject Classification

  • 18E30 (primary);
  • 16E99;
  • 16G70;
  • 18D99;
  • 18G99 (secondary)

Keywords

  • preprojective algebras;
  • 2-Calabi–Yau categories;
  • reduced expressions;
  • quiver mutation;
  • cluster algebras;
  • cluster categories;
  • cluster tilting;
  • AR-quivers

Footnotes

The authors were supported by a STORFORSK grant (no. 167130) from the Norwegian Research Council.