×

Companion bases for cluster-tilted algebras. (English) Zbl 1336.13012

Cluster algebras were introduced in [S. Fomin and A. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)]. Those of finite type are classified by the Dynkin diagrams [S. Fomin and A. Zelevinsky, Invent. Math. 154, No. 1, 63–121 (2003; Zbl 1054.17024)]. A cluster algebra consists of a collection of seeds, each containing a quiver and a free generating set of a fixed field of rational functions in finitely many variables. Given one such seed, the others can be obtained via mutation. Given a cluster algebra of finite type with associated Dynkin diagram which is simply-laced, the article under review introduces the notion of a companion basis for a seed in the cluster algebra. Such a basis is a \(\mathbb{Z}\)-basis (consisting of roots) of the integral root lattice of the corresponding root system indexed by the vertices in the quiver of the seed. It has the property that the absolute value of the inner product of a pair of distinct roots in the basis is \(1\) if there is an arrow between the corresponding vertices in the quiver, and zero otherwise.
The article explains the relationship between the different companion bases for a fixed seed. It also shows, in the type \(A\) case, that the dimension vectors of the finite dimensional indecomposable modules over a cluster-tilted algebra may be found, up to sign, by writing each of the positive roots in terms of a companion basis for a seed containing the quiver of the algebra (noting that such a seed necessarily exists). This can be regarded as a generalization of part of Gabriel’s Theorem. Finally, it is shown (in the simply-laced Dynkin case) how a companion basis for the mutation of a quiver can be obtained from a companion basis for the original quiver.

MSC:

13F60 Cluster algebras
16G10 Representations of associative Artinian rings
16G20 Representations of quivers and partially ordered sets
05E10 Combinatorial aspects of representation theory
18E30 Derived categories, triangulated categories (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Assem, I., Brüstle, T., Charbonneau-Jodoin, G., Plamondon, P-G.: Gentle algebras arising from surface triangulations. Algebra and Number Theory 4(2), 201-229 (2010) · Zbl 1242.16011 · doi:10.2140/ant.2010.4.201
[2] Auslander, M., Platzeck, M.I., Reiten, I.: Coxeter functors without diagrams. Trans. Am. Math. Soc. 250, 1-46 (1979) · Zbl 0421.16016 · doi:10.1090/S0002-9947-1979-0530043-2
[3] Barot, M., Geiss, C., Zelevinsky, A.: Cluster algebras of finite type and positive symmetrizable matrices. J. Lond. Math. Soc. 73, 545-564 (2006) · Zbl 1093.05070 · doi:10.1112/S0024610706022769
[4] Barot, M., Marsh, R.: Reflection group presentations arising from cluster algebras. Preprint arXiv:1112.2300 · Zbl 1444.20026
[5] Bourbaki, N.: Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4-6. Springer (2002) · Zbl 0983.17001
[6] Buan, A., Marsh, R., Reineke, M., Reiten, I., Todorov, G.: Tilting theory and cluster combinatorics. Adv. Math. 204(2), 572-618 (2006) · Zbl 1127.16011 · doi:10.1016/j.aim.2005.06.003
[7] Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras. Trans. Am. Math. Soc. 359(1), 323-332 (2007) · Zbl 1123.16009 · doi:10.1090/S0002-9947-06-03879-7
[8] Buan, A., Marsh, R., Reiten, I.: Cluster mutation via quiver representations. Comment. Math. Helv. 83(2), 143-177 (2008) · Zbl 1193.16016 · doi:10.4171/CMH/121
[9] Buan, A., Marsh, R., Reiten, I.: Cluster-tilted algebras of finite representation type. J. Algebra 306(2), 412-431 (2006) · Zbl 1116.16012 · doi:10.1016/j.jalgebra.2006.08.005
[10] Buan, A., Vatne, D.: Derived equivalence classification for cluster-tilted algebras of type An. J. Algebra 319(7), 2723-2738 (2008) · Zbl 1155.16010 · doi:10.1016/j.jalgebra.2008.01.007
[11] Butler, M.C.R., Ringel, C.M.: Auslander-Reiten sequences with few middle terms and applications to string algebras. Comm. Algebra 15, 145-179 (1987) · Zbl 0612.16013 · doi:10.1080/00927878708823416
[12] Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations arising from clusters (An case). Trans. Am. Math. Soc. 358(3), 1347-1364 (2006) · Zbl 1137.16020 · doi:10.1090/S0002-9947-05-03753-0
[13] Caldero, P., Chapoton, F., Schiffler, R.: Quivers with relations and cluster-tilted algebras. Algebr. Represent. Theory 9(4), 359-376 (2006) · Zbl 1127.16013 · doi:10.1007/s10468-006-9018-1
[14] Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Springer (1999) · Zbl 0915.52003
[15] Fomin, S., Zelevinsky, A.: Cluster algebras I: foundations. J. Am. Math. Soc. 15(2), 497-529 (2002) · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X
[16] Fomin, S., Zelevinsky, A.: Cluster algebras II: finite type classification. Invent. Math 154(1), 63-121 (2003) · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y
[17] Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. 158(3), 977-1018 (2003) · Zbl 1057.52003 · doi:10.4007/annals.2003.158.977
[18] Gabriel, P.: Unzerlegbare darstellungen I. Manuscr. Math. 6, 71-103 (1972) · Zbl 0232.08001 · doi:10.1007/BF01298413
[19] Humphreys, J.E.: Reflection groups and coxeter groups. In: Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press (1990) · Zbl 0725.20028
[20] Marsh, R., Reineke, M., Zelevinsky, A.: Generalized associahedra via quiver representations. Trans. Am. Math. Soc. 355(10), 4171-4186 (2003) · Zbl 1042.52007 · doi:10.1090/S0002-9947-03-03320-8
[21] Parsons, M.J.: On indecomposable modules over cluster-tilted algebras of type A. Ph.D. Thesis, University of Leicester (2007) · Zbl 1242.16011
[22] Reiten, I.: Tilting theory and cluster algebras. Preprint arXiv:1012.6014 · Zbl 0906.16002
[23] Ringel, C.M.: Tame algebras and integral quadratic forms. In: Lecture Notes in Mathematics, vol. 1099. Springer (1984) · Zbl 0546.16013
[24] Ringel, C.M.: Cluster-concealed algebras. Adv. Math. 226(2), 1513-1537 (2011) · Zbl 1238.16017 · doi:10.1016/j.aim.2010.08.014
[25] Samelson, H.: Notes on Lie Algebras. Revised Edition, Springer (1990) · Zbl 0708.17005
[26] Seven, A.: Recognizing cluster algebras of finite type. Electron. J. Combin. 14(1), Research Paper 3 (2007) · Zbl 1114.05103
[27] Seven, A.: Reflection group relations arising from cluster algebras. Preprint arXiv:1210.6217 · Zbl 1345.05114
[28] Zhu, B.: BGP-reflection functors and cluster combinatorics. J. Pure Appl. Algebra 209(2), 497-506 (2007) · Zbl 1127.16014 · doi:10.1016/j.jpaa.2006.06.006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.