Skip Navigation


Journal of the London Mathematical Society Advance Access originally published online on July 21, 2007
Journal of the London Mathematical Society 2007 75(3):718-740; doi:10.1112/jlms/jdm017
This Article
Right arrow Full Text (PDF)
Right arrow All Versions of this Article:
75/3/718    most recent
jdm017v2
jdm017v1
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
Google Scholar
Right arrow Articles by Geiß, C.
Right arrow Articles by Schröer, J.
Right arrow Search for Related Content
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© 2007 London Mathematical Society

Auslander algebras and initial seeds for cluster algebras

Christof Geiß1,, Bernard Leclerc3 and Jan Schröer2

1 Instituto de Matemáticas
Universidad Nacional Autónoma de México
México D.F.
C.P. 04510
México

2 Mathematisches Institut
Universität Bonn
Beringstr. 1
53115 Bonn
Germany
jschroer{at}maths.leeds.ac.uk

3 Université de Caen
LMNO CNRS UMR 6139
F-14032 Caen cedex
France
leclerc{at}math.unicaen.fr

Let Q be a Dynkin quiver and {Pi} the corresponding set of positive roots. For the preprojective algebra {Lambda} associated to Q, a rigid {Lambda}-module IQ is produced with r = |{Pi}| pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of k Q to {Lambda}. If N is a maximal unipotent subgroup of a complex simply connected simple Lie group of type |Q|, then the coordinate ring C[N] is an upper cluster algebra. It is shown that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of IQ coincide with certain generalized minors which form an initial cluster for C[N] and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of End{Lambda}(IQ). Finally, the fact that the categories of injective modules over {Lambda} and over its covering Formula are triangulated is exploited in order to show several interesting identities in the respective stable module categories.

christof{at}matem.unam.mx

2000 Mathematics Subject Classification 14M99, 16D70, 16E20, 16G20, 16G70, 17B37, 20G42.

The first author acknowledges support from DGAPA grant IN101402-3. The second author is grateful to the CNRS networks GDR 2432 and the GDR 2249 for their support. The third author was supported by a research fellowship from the DFG.

Received September 2, 2005; revised August 24, 2006; published online July 21, 2007.


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.