Skip Navigation

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn029, 17 pages, doi:10.1093/imrn/rnn029 published on April 25, 2008
This Article
Right arrow Full Text (PDF)
Right arrow References
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Dehy, R.
Right arrow Articles by Keller, B.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

On the Combinatorics of Rigid Objects in 2–Calabi–Yau Categories

Raika Dehy1 and Bernhard Keller2

1 Universitée Cergy–Pontoise/Saint–Martin, Déepartment de Mathéematiques, UMR 8088 du CNRS, 2 avenue Adolphe Chauvin, 95302 Cergy–Pontoise Cedex, France
2 UFR de Mathéematiques, Universitée Denis Diderot – Paris 7, Institut de Mathéematiques, UMR 7586 du CNRS, 2, place Jussieu, 75251 Paris Cedex 05, France

Correspondence: Correspondence to be sent to: keller{at}math.jussieu.fr

Given a triangulated 2-Calabi–Yau category C and a cluster-tilting subcategory T, the index of an object X of C is a certain element of the Grothendieck group of the additive category T. In this note, we show that a rigid object of C is determined by its index, that the indices of the indecomposables of a cluster-tilting subcategory T' form a basis of the Grothendieck group of T and that, if T and T' are related by a mutation, then the indices with respect to T and T' are related by a certain piecewise linear transformation introduced by Fomin and Zelevinsky in their study of cluster algebras with coefficients. This allows us to give a combinatorial construction of the indices of all rigid objects reachable from the given cluster-tilting subcategory T. Conjecturally, these indices coincide with Fomin–Zelevinsky's g-vectors.


Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?




Disclaimer:
Please note that abstracts for content published before 1996 were created through digital scanning and may therefore not exactly replicate the text of the original print issues. All efforts have been made to ensure accuracy, but the Publisher will not be held responsible for any remaining inaccuracies. If you require any further clarification, please contact our Customer Services Department.