Abstract
For any given symmetrizable Cartan matrix C with a symmetrizer D, Geiß et al. (2016) introduced a generalized preprojective algebra Π(C, D). We study tilting modules and support τ-tilting modules for the generalized preprojective algebra Π(C, D) and show that there is a bijection between the set of all cofinite tilting ideals of Π(C, D) and the corresponding Weyl group W(C) provided that C has no component of Dynkin type. When C is of Dynkin type, we also establish a bijection between the set of all basic support τ-tilting Π(C, D)-modules and the corresponding Weyl group W(C). These results generalize the classification results of Buan et al. (Compos. Math. 145(4), 1035–1079 2009) and Mizuno (Math. Zeit. 277(3), 665–690 2014) over classical preprojective algebras.
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References
Adachi, T., Iyama, O., Reiten, I.: τ-tilting theory. Compos. Math. 150(3), 415–452 (2014)
Björner, A., Brenti, F.: Combinatorics of Coxeter Groups, Graduate Texts in Mathematics, vol. 231. Springer, Berlin (2005)
Brenner, S., Butler, M., King, A.: Periodic algebras which are almost Koszul. Algebras Represent. Theory 5, 331–367 (2002)
Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145(4), 1035–1079 (2009)
Carter, R.W.: Lie Algebras of Finite and Affine Type, Cambridge Studies in Advanced Mathematics. Cambridge University Press (2005)
Crawley-Boevey, W.: On the exceptional fibres of Kleinian singularities. Amer. J. Math. 122, 1027–1037 (2000)
Gelfand, I.M., Ponomare, V.A.: Model algebras and representations of graphs. Funktsional. Anal. i Prilozhen. 13, 1–12 (1979)
Iyama, O., Reiten, I.: Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras. Amer. J. Math. 130, 1089–1149 (2008)
Lusztig, G.: Semicanonical bases arising from enveloping algebras. Adv. Math. 151, 129–139 (2000)
Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. 76(2), 365–416 (1994)
Geiß, C., Leclerc, B., Schröer, J.: Semicanonical bases and preprojective algebras. Ann. Sc. École Norm. Sup. 38, 193–253 (2005)
Geiß, C., Leclerc, B., Schröer, J.: Rigid modules over preprojective algebras. Invent. Math. 165, 589–632 (2006)
Geiß, C., Leclerc, B., Schröer, J.: Kac-Moody groups and cluster algebras. Adv. Math. 228, 329–433 (2011)
Geiß, C., Leclerc, B., Schröer, J.: Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. https://doi.org/10.1007/s00222-016-0705-1 (2016)
Mizuno, Y.: Classifying τ-tilting modules over preprojective algebras of Dynkin type. Math. Zeit. 277(3), 665–690 (2014)
Rickard, J.: Morita theory for derived categories. J. London Math. Soc. 29(2), 436–456 (1989)
Ringel, C.M.: The preprojective algebra of a quiver. In: Algebras and Modules II (Geiranger, 1966), pp. 467–480, CMS Conf. Proc. 24. AMS (1998)
Yekutieli, A.: Dualizing complexes, Morita equivalence and the derived Picard group of a ring. J. London Math. Soc. 60(2), 723–746 (1999)
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Partially supported by the China Scholarship Council and the National Natural Science Foundation of China (No. 11471224).
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Presented by: Michel Van den Bergh
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Fu, C., Geng, S. Tilting Modules and Support τ-Tilting Modules over Preprojective Algebras Associated with Symmetrizable Cartan Matrices. Algebr Represent Theor 22, 1239–1260 (2019). https://doi.org/10.1007/s10468-018-9819-z
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DOI: https://doi.org/10.1007/s10468-018-9819-z
Keywords
- Symmetrizable Cartan matrix
- Preprojective algebras
- Locally free modules
- Generalized simple modules
- Cofinite tilting ideals
- Support τ-tilting modules