Partial flag varieties and preprojective algebras
Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 825-876.

Let Λ be a preprojective algebra of type A,D,E, and let G be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories Sub Q for Q an injective Λ-module, and we introduce a mutation operation between complete rigid modules in Sub Q. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to G.

Soit Λ une algèbre préprojective de type A,D,E, et soit G le groupe algébrique complexe semi-simple et simplement connexe correspondant. Nous étudions les modules rigides des sous-catégories Sub QQ désigne un Λ-module injectif, et nous introduisons une opération de mutation sur les modules rigides complets de Sub Q. Ceci conduit à des structures d’algèbre amassée sur les anneaux de coordonnées des variétés de drapeaux partiels associées à G.

DOI: 10.5802/aif.2371
Classification: 14M15,  16D90,  16G20,  16G70,  17B10,  20G05,  20G20,  20G42
Keywords: Flag variety, preprojective algebra, Frobenius category, rigid module, mutation, cluster algebra, semicanonical basis
Geiß, Christof 1; Leclerc, Bernard 2; Schröer, Jan 3

1 Universidad Nacional Autónoma de México Instituto de Matemáticas 04510 México D.F. (México)
2 Université de Caen LMNO UMR 6139 14032 Caen cedex (France)
3 Universität Bonn Mathematisches Institut Beringstr. 1 53115 Bonn (Germany)
@article{AIF_2008__58_3_825_0,
     author = {Gei{\ss}, Christof and Leclerc, Bernard and Schr\"oer, Jan},
     title = {Partial flag varieties and preprojective algebras},
     journal = {Annales de l'Institut Fourier},
     pages = {825--876},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {58},
     number = {3},
     year = {2008},
     doi = {10.5802/aif.2371},
     mrnumber = {2427512},
     zbl = {1151.16009},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2371/}
}
TY  - JOUR
AU  - Geiß, Christof
AU  - Leclerc, Bernard
AU  - Schröer, Jan
TI  - Partial flag varieties and preprojective algebras
JO  - Annales de l'Institut Fourier
PY  - 2008
DA  - 2008///
SP  - 825
EP  - 876
VL  - 58
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2371/
UR  - https://www.ams.org/mathscinet-getitem?mr=2427512
UR  - https://zbmath.org/?q=an%3A1151.16009
UR  - https://doi.org/10.5802/aif.2371
DO  - 10.5802/aif.2371
LA  - en
ID  - AIF_2008__58_3_825_0
ER  - 
%0 Journal Article
%A Geiß, Christof
%A Leclerc, Bernard
%A Schröer, Jan
%T Partial flag varieties and preprojective algebras
%J Annales de l'Institut Fourier
%D 2008
%P 825-876
%V 58
%N 3
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2371
%R 10.5802/aif.2371
%G en
%F AIF_2008__58_3_825_0
Geiß, Christof; Leclerc, Bernard; Schröer, Jan. Partial flag varieties and preprojective algebras. Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 825-876. doi : 10.5802/aif.2371. http://archive.numdam.org/articles/10.5802/aif.2371/

[1] Auslander, M.; Smalø, S. Almost split sequences in subcategories, J. Algebra, Volume 69 (1981), pp. 426-454 | DOI | MR | Zbl

[2] Auslander, Maurice; Reiten, Idun Homologically finite subcategories, Representations of algebras and related topics (Kyoto, 1990) (London Math. Soc. Lecture Note Ser.), Volume 168, Cambridge Univ. Press, 1992, pp. 1-42 | MR | Zbl

[3] Bautista, R.; Martinez, R. Representations of partially ordered sets and 1-Gorenstein Artin algebras, Proceedings, Conference on Ring Theory, Antwerp, 1978 (1979), pp. 385-433 | MR | Zbl

[4] Berenstein, A.; Fomin, S.; Zelevinsky, A. Cluster algebras III. Upper bounds and double Bruhat cells, Duke Math. J., Volume 126 (2005), pp. 1-52 | DOI | MR

[5] Borel, A. Linear algebraic groups, 2nd Enlarged Edition, Springer, 1991 | MR | Zbl

[6] Bourbaki, N. Groupes et algèbres de Lie, chap 4, 5, 6, Hermann, 1968 | MR | Zbl

[7] Buan, A.; Iyama, O.; Reiten, I.; Scott, J. Cluster structures for 2-Calabi-Yau categories and unipotent groups (arXiv:math.RT/0701557)

[8] Buan, A.; Marsh, R.; Reineke, M.; Reiten, I.; Todorov, G. Tilting theory and cluster combinatorics, Adv. Math., Volume 204 (2006), pp. 572-618 | DOI | MR | Zbl

[9] Fomin, S.; Zelevinsky, A. Double Bruhat cells and total positivity, J. Amer. Math. Soc., Volume 12 (1999), pp. 335-380 | DOI | MR | Zbl

[10] Fomin, S.; Zelevinsky, A. Cluster algebras. I. Foundations, J. Amer. Math. Soc., Volume 15 (2002), pp. 497-529 | DOI | MR | Zbl

[11] Fomin, S.; Zelevinsky, A. Cluster algebras II. Finite type classification, Invent. Math., Volume 154 (2003), pp. 63-121 | DOI | MR | Zbl

[12] Geiß, C.; Leclerc, B.; Schröer, J. Cluster algebra structures and semicanonical bases for unipotent subgroups (arXiv:math.RT/0703039)

[13] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical bases and preprojective algebras, Ann. Sci. Ecole Norm. Sup., Volume 38 (2005), pp. 193-253 | Numdam | MR

[14] Geiß, C.; Leclerc, B.; Schröer, J. Rigid modules over preprojective algebras, Invent. Math., Volume 165 (2006), pp. 589-632 | DOI | MR

[15] Geiß, C.; Leclerc, B.; Schröer, J. Verma modules and preprojective algebras, Nagoya Math. J., Volume 182 (2006), pp. 241-258 | MR | Zbl

[16] Geiß, C.; Leclerc, B.; Schröer, J. Auslander algebras and initial seeds for cluster algebras, J. London Math. Soc., Volume 75 (2007), pp. 718-740 | DOI | MR

[17] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical bases and preprojective algebras II: A multiplication formula, Compositio Math., Volume 143 (2007), pp. 1313-1334 | DOI | MR

[18] Geiß, C.; Schröer, J. Extension-orthogonal components of preprojective varieties, Trans. Amer. Math. Soc., Volume 357 (2005), pp. 1953-1962 | DOI | MR | Zbl

[19] Gekhtman, M.; Shapiro, M.; Vainshtein, A. Cluster algebras and Poisson geometry, Moscow Math. J., Volume 3 (2003), pp. 899-934 | MR | Zbl

[20] Happel, D. Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series 119, Cambridge University Press, Cambridge, 1988 (x+208pp) | MR | Zbl

[21] Hoshino, M. On splitting torsion theories induced by tilting modules, Comm. Alg., Volume 11 (1983), pp. 427-439 | DOI | MR | Zbl

[22] Kashiwara, M.; Saito, Y. Geometric construction of crystal bases, Duke Math. J., Volume 89 (1997), pp. 9-36 | DOI | MR | Zbl

[23] Keller, B. On triangulated orbit categories, Doc. Math., Volume 10 (2005), pp. 551-581 (electronic) | MR | Zbl

[24] Kleiner, M. Approximations and almost split sequences in homologically finite subcategories, J. Algebra, Volume 198 (1997), pp. 135-163 | DOI | MR | Zbl

[25] Lakshmibai, V.; Gonciulea, N. Flag varieties, Travaux en cours, 63, Hermann, 2001 | Zbl

[26] Lusztig, G. Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., Volume 3 (1990), pp. 447-498 | DOI | MR | Zbl

[27] Lusztig, G. Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc., Volume 4 (1991), pp. 365-421 | DOI | MR | Zbl

[28] Lusztig, G. Introduction to quantum groups, Birkhäuser, 1993 | MR | Zbl

[29] Postnikov, A. Total positivity, grassmannians and networks (arXiv:math.CO/0609764.)

[30] Ringel, M. C. The preprojective algebra of a quiver, Algebras and Modules II, (Geiranger, 1996), CMS Conf. Proc., Volume 24 (1998), pp. 467-480 (AMS) | MR | Zbl

[31] Scott, J. Grassmannians and cluster algebras, Proc. London Math. Soc., Volume 92 (2006), pp. 345-380 | DOI | MR | Zbl

[32] Seven, Ahmet I. Recognizing cluster algebras of finite type, Electron. J. Combin., Volume 14 (2007) no. 1, pp. Research Paper 3, 35 pp. (electronic) | MR | Zbl

Cited by Sources: