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On the module categories of generalized preprojective algebras of Dynkin type. (English) Zbl 1510.16008

Summary: For a symmetrizable GCM \(C\) and its symmetrizer \(D\), Geiss-Leclerc-Schröer [C. Geiss et al., Invent. Math. 209, No. 1, 61–158 (2017; Zbl 1395.16006)] has introduced a generalized preprojective algebra \(\Pi\) associated to \(C\) and \(D\), that contains a class of modules, called locally free modules. We show that any basic support \(\tau \)-tilting \(\Pi \)-module is locally free and gives a classification theorem of torsion-free classes in \(\operatorname{\mathbf{rep}}{\Pi}\) as the generalization of the work of Y. Mizuno [Math. Z. 277, No. 3–4, 665–690 (2014; Zbl 1355.16008)].

MSC:

16G10 Representations of associative Artinian rings
16G20 Representations of quivers and partially ordered sets
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References:

[1] T. Adachi, O. Iyama and I. Reiten: \( \tau \)-tilting theory, Compos. Math. 150 (2014), 415-452. · Zbl 1330.16004
[2] S. Asai: Bricks over preprojective algebras and join-irreducible elements in coxeter groups, 2017, arXiv:1712.08311.
[3] I. Assem, D. Simson and A. Skowroński: Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts 65, Cambridge University Press, Cambridge, 2006. · Zbl 1092.16001
[4] P. Baumann, J. Kamnitzer and P. Tingley: Affine Mirković-Vilonen polytopes, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 113-205. · Zbl 1332.17012
[5] A. Björner and F. Brenti: Combinatorics of Coxeter groups, Graduate Texts in Mathematics 231, Springer, New York, 2005. · Zbl 1110.05001
[6] S. Brenner and M.C.R. Butler: Generalizations of the Bernstein-Gel’fand-Ponomarev reflection functors; in Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math. 832, Springer, Berlin-New York, 1980, 103-169. · Zbl 0446.16031
[7] A.B. Buan, O. Iyama, I. Reiten and J. Scott: Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math. 145 (2009), 1035-1079. · Zbl 1181.18006
[8] R.W. Carter: Lie algebras of finite and affine type, Cambridge Studies in Advanced Mathematics 96, Cambridge University Press, Cambridge, 2005. · Zbl 1110.17001
[9] L. Demonet, O. Iyama and G. Jasso: \( \tau \)-Tilting Finite Algebras, Bricks, and \(g\)-Vectors, Int. Math. Res. Not. IMRN 2019 (2019), 852-892. · Zbl 1485.16013
[10] C. Fu and S. Geng: Tilting modules and support \(\tau \)-tilting modules over preprojective algebras associated with symmetrizable Cartan matrices, Algebr. Represent. Theory 22 (2019), 1239-1260. · Zbl 1454.16012
[11] P. Gabriel: Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71-103; correction, ibid. 6 (1972), 309. · Zbl 0232.08001
[12] C. Geiß, B. Leclerc and J. Schröer: Kac-Moody groups and cluster algebras, Adv. Math. 228 (2011), 329-433. · Zbl 1232.17035
[13] C. Geiss, B. Leclerc and J. Schröer: Quivers with relations for symmetrizable Cartan matrices I: Foundations, Invent. Math. 209 (2017), 61-158. · Zbl 1395.16006
[14] I.M. Gelfand and V.A. Ponomarev: Model algebras and representations of graphs, Funktsional. Anal. I Prilozhen. 13 (1979), 1-12. · Zbl 0437.16020
[15] D. Happel: Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, 119, Cambridge University Press, Cambridge, 1988. · Zbl 0635.16017
[16] O. Iyama, N. Reading, I. Reiten and H. Thomas: Lattice structure of Weyl groups via representation theory of preprojective algebras, Compos. Math. 154 (2018), 1269-1305. · Zbl 1443.16016
[17] V.G. Kac: Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57-92. · Zbl 0427.17001
[18] V.G. Kac: Infinite root systems, representations of graphs and invariant theory. II, J. Algebra 78 (1982), 141-162. · Zbl 0497.17007
[19] Y. Kimura: Tilting theory of preprojective algebras and c-sortable elements, J. Algebra 503 (2018), 186-221. · Zbl 1432.16013
[20] Y. Miyashita: Tilting modules of finite projective dimension, Math. Z. 193 (1986), 113-146. · Zbl 0578.16015
[21] Y. Mizuno: Classifying \(\tau \)-tilting modules over preprojective algebras of Dynkin type, Math. Z. 277 (2014), 665-690. · Zbl 1355.16008
[22] S. Oppermann, I. Reiten and H. Thomas: Quotient closed subcategories of quiver representations, Compos. Math. 151 (2015), 568-602. · Zbl 1366.16010
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