|
Quivers with relations arising from clusters  case)
Author(s):
P.
Caldero;
F.
Chapoton;
R.
Schiffler
Journal:
Trans. Amer. Math. Soc.
358
(2006),
1347-1364.
MSC (2000):
Primary 16G20, 16G70, 05E15, 20F55
Posted:
May 26, 2005
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
Abstract:
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let  be a cluster algebra of type  . We associate to each cluster  of  an abelian category  such that the indecomposable objects of  are in natural correspondence with the cluster variables of  which are not in  . We give an algebraic realization and a geometric realization of  . Then, we generalize the ``denominator theorem'' of Fomin and Zelevinsky to any cluster.
References:
-
- 1.
- M. Auslander, I. Reiten, and S. O. Smalø, Representation theory of Artin algebras, volume 36 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995. MR 1476671 (98e:16011)
- 2.
- A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. Journal, 126 (1), 1-52, 2005. MR 2110627
- 3.
- A. Buan, R. J. Marsh, M. Reineke, I. Reiten, and G. Todorov, Tilting theory and cluster combinatorics, arXiv:math.RT/0402054.
- 4.
- A. Buan, R. J. Marsh, and I. Reiten, Cluster-tilted algebras, arXiv:math.RT/0402075.
- 5.
- S. Fomin and A. Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15 (2), 497-529, 2002. MR 1887642 (2003F:16050)
- 6.
- S. Fomin and A. Zelevinsky, Cluster algebras. II. Finite type classification, Inventiones Mathematicae, 154 63-121, 2003. MR 2004457 (2004m:17011)
- 7.
- S. Fomin and A. Zelevinsky,
 -systems and generalized associahedra, Ann. of Math. (2), 158 (3), 977-1018, 2003. MR 2031858 (2004m:17010) - 8.
- P. Gabriel, Auslander-Reiten sequences and representation-finite algebras, in Representation Theory, I (Proc. Workshop, Carleton Univ., Ottawa, Ont., 1979), volume 831 of Lecture Notes in Math., pages 1-71, Springer, Berlin, 1980. MR 0607140 (82i:16030)
- 9.
- B. Keller, Triangulated orbit categories, Preprint, 2003.
- 10.
- R. Marsh, M. Reineke, and A. Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc., 355 (10), 4171-4186, 2003. MR 1990581 (82i:16030)
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
16G20, 16G70, 05E15, 20F55
Retrieve articles in all Journals with MSC
(2000):
16G20, 16G70, 05E15, 20F55
Additional Information:
P.
Caldero
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France
F.
Chapoton
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon 1, 69622 Villeurbanne Cedex, France
R.
Schiffler
Affiliation:
School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6
DOI:
10.1090/S0002-9947-05-03753-0
PII:
S 0002-9947(05)03753-0
Received by editor(s):
February 25, 2004
Received by editor(s) in revised form:
May 24, 2004
Posted:
May 26, 2005
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
|
Comments: webmaster@ams.org