Algèbre
Algèbres amassées et algèbres préprojectives : le cas non simplement lacé
[Cluster algebras and preprojective algebras: the non simply-laced case]
Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 379-384.

We generalize to the non simply-laced case results of Geiß, Leclerc and Schröer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case.

On généralise au cas non simplement lacé des résultats de Geiß, Leclerc et Schröer sur les structures amassées des algèbres de fonctions sur les sous-groupes unipotents maximaux des groupes de Lie simples. Cela permet en particulier de voir les structures amassées dans le cas non simplement lacé comme projections des structures amassées dans le cas simplement lacé. Cela permet aussi de montrer la liberté des monômes d'amas dans le cas non simplement lacé.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.02.007
Demonet, Laurent 1

1 LMNO, université de Caen, esplanade de la Paix, 14000 Caen, France
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Demonet, Laurent. Algèbres amassées et algèbres préprojectives : le cas non simplement lacé. Comptes Rendus. Mathématique, Volume 346 (2008) no. 7-8, pp. 379-384. doi : 10.1016/j.crma.2008.02.007. http://www.numdam.org/articles/10.1016/j.crma.2008.02.007/

[1] Berenstein, A.; Fomin, S.; Zelevinsky, A. Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J., Volume 126 (2005) no. 1, pp. 1-52

[2] Dupont, G. An approach to non simply laced cluster algebras | arXiv

[3] Fomin, S.; Zelevinsky, A. Cluster algebras I: Foundations, J. Amer. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529

[4] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical bases and preprojective algebras, Ann. Sci. École Norm. Sup. (4), Volume 38 (2005) no. 2, pp. 193-253

[5] Geiß, C.; Leclerc, B.; Schröer, J. Rigid modules over preprojective algebras, Invent. Math., Volume 165 (2006) no. 3, pp. 589-632

[6] Geiß, C.; Leclerc, B.; Schröer, J. Semicanonical bases and preprojective algebras II: A multiplication formula, Compositio Math., Volume 143 (2007) no. 5, pp. 1313-1334

[7] Geiß, C.; Leclerc, B.; Schröer, J. Partial flag varieties and preprojective algebras (Ann. Inst. Fourier, à paraître) | arXiv

[8] Kac, V.G. Infinite Dimensional Lie Algebras, Cambridge University Press, 1994

[9] Lusztig, G. Semicanonical bases arising from enveloping algebras, Adv. Math., Volume 151 (2000) no. 2, pp. 129-139

[10] Lusztig, G. Introduction to Quantum Groups, Progress in Mathematics, vol. 110, Birkhäuser, Boston, 1993

[11] Reiten, I.; Riedtmann, C. Skew group algebras in the representation theory of Artin algebras, J. Algebra, Volume 92 (1985) no. 1, pp. 224-282

[12] Yang, D. Non-simply-laced clusters of finite type via Frobenius morphism | arXiv

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