Classifying \(\tau \) -tilting modules over preprojective algebras of Dynkin type
详细信息 查看全文
- 作者：Yuya Mizuno (1)
- 刊名：Mathematische Zeitschrift
- 出版年：2014
- 出版时间：August 2014
- 年：2014
- 卷：277
- 期：3-4
- 页码：665-690
- 全文大小：357 KB
- 参考文献：1. Adachi, T., Iyama, O., Reiten, I.: \(\tau \) -tilting theory, arXiv: 1210.1036 (to appear in Compos. Math., 2012)
2. Amiot, C., Iyama, O., Reiten, I., Todorov, G.: Preprojective algebras and c-sortable words. Proc. Lond. Math. Soc. (3) 104(3), 513鈥?39 (2012) CrossRef
3. Assem, I., Simson, D., Skowro艅ski, A.: Elements of the Representation Theory of Associative Algebras. Vol. 1, London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006) CrossRef
4. Auslander, M., Reiten, I.: Modules determined by their composition factors, Ill. J. Math. 29, 280鈥?01 (1985)
5. Auslander, M., Reiten, I.: \(DTr\) -periodic modules and functors. Representation theory of algebras (Cocoyoc, 1994), pp. 39鈥?0. In: CMS Conference Proceedings, 18, American Mathematical Society, Providence, RI (1996)
6. Auslander, M., Reiten, I., Smal酶, S.O.: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics, vol. 36. Cambridge University Press, Cambridge (1995) CrossRef
7. Baer, D., Geigle, W., Lenzing, H.: The preprojective algebra of a tame hereditary Artin algebra. Commun. Algebra 15(1鈥?), 425鈥?57 (1987) CrossRef
8. Baumann, P., Kamnitzer, J., Tingley, P.: Affine Mirkovi膰鈥揤ilonen polytopes, arXiv:1110.3661 (to appear in Publ. IHES)
9. Bj枚rner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York (2005)
10. Bocklandt, R.: Graded Calabi Yau algebras of dimension 3. J. Pure Appl. Algebra 212(1), 14鈥?2 (2008) CrossRef
11. Bourbaki, N.: Lie groups and Lie algebras. Chapters 4鈥?, Elements of Mathematics (Berlin), transl. from the 1968 French original by Andrew Pressley. Springer, Berlin (2002)
12. Brenner, S., Butler, M.C.R., King, A.D.: Periodic algebras which are Almost Koszul. Algebr. Represent. Theory 5(4), 331鈥?67 (2002) CrossRef
13. Br眉stle, T., Yang, D.: Ordered exchange graphs, arXiv:1302.6045 (to appear in Adv. Represent. Theory Algebras (ICRA Bielefeld 2012))
14. Buan, A.B., Iyama, O., Reiten, I., Scott, J.: Cluster structures for 2-Calabi鈥揧au categories and unipotent groups. Compos. Math. 145, 1035鈥?079 (2009) CrossRef
15. Crawley-Boevey, W.: On the exceptional fibres of Kleinian singularities. Am. J. Math. 122(5), 1027鈥?037 (2000) CrossRef
16. Dehy, R., Keller, B.: On the combinatorics of rigid objects in 2-Calabi鈥揧au categories. Int. Math. Res. Not. IMRN 2008, no. 11, Art. ID rnn029
17. Dlab, V., Ringel, C.M.: The preprojective algebra of a modulated graph, Representation theory, II. In: Proceedings of Second International Conference on Carleton University, Ottawa, Ontario, 1979, pp. 216鈥?31, Lecture Notes in Mathematics, vol. 832. Springer, Berlin, New York (1980)
18. Dlab, V., Ringel, C.M.: The module theoretical approach to quasi-hereditary algebras. Representations of algebras and related topics (Kyoto, 1990), pp. 200鈥?24, London Mathematical Society Lecture Note Series, 168, Cambridge University Press, Cambridge (1992)
19. Geiss, C., Leclerc, B., Schr枚er, J.: Rigid modules over preprojective algebras. Invent. Math. 165(3), 589鈥?32 (2006) CrossRef
20. Geiss, C., Leclerc, B., Schr枚er, J.: Semicanonical bases and preprojective algebras II: a multiplication formula. Compos. Math. 143(5), 1313鈥?334 (2007) CrossRef
21. Geiss, C., Leclerc, B., Schr枚er, J.: Kac鈥揗oody groups and cluster algebras. Adv. Math. 228(1), 329鈥?33 (2011) CrossRef
22. Gelfand, I.M., Ponomarev, V.A.: Model algebras and representations of graphs. Funktsional. Anal. i Prilozhen. 13(3), 1鈥?2 (1979)
23. Humphreys, J.E.: Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, vol. 29. Cambridge University Press, Cambridge (1990) CrossRef
24. Ingalls, C., Thomas, H.: Noncrossing partitions and representations of quivers. Compos. Math. 145(6), 1533鈥?562 (2009) CrossRef
25. Iyama, O., Jasso, G.: On \(g\) -vectors of \(\tau \) -tilting modules and \(\tau \) -rigid-finite algebras (in preparation)
26. Iyama, O., Reiten, I.: Fomin鈥揨elevinsky mutation and tilting modules over Calabi鈥揧au algebras. Am. J. Math. 130(4), 1087鈥?149 (2008) CrossRef
27. Koenig, S., Yang, D.: Silting objects, simple-minded collections, \(t\) -structures and co- \(t\) -structures for finite-dimensional algebras. arXiv:1203.5657
28. Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4(2), 365鈥?21 (1991) CrossRef
29. Lusztig, G.: Semicanonical bases arising from enveloping algebras. Adv. Math. 151(2), 129鈥?39 (2000) CrossRef
30. Mizuno, Y.: \(\nu \) -stable \(\tau \) -tilting modules, arXiv:1210.8322 (to appear in Commun. Algebra, 2012)
31. Oppermann, S., Reiten, I., Thomas, H.: Quotient closed subcategories of quiver representations, arXiv:1205.3268 (2012)
32. Palu, Y.: Cluster characters for 2-Calabi鈥揧au triangulated categories. Ann. Inst. Fourier (Grenoble) 58(6), 2221鈥?248 (2008) CrossRef
33. Reading, N.: Clusters, coxeter-sortable elements and noncrossing partitions. Trans. Am. Math. Soc. 359(12), 5931鈥?958 (2007) CrossRef
34. Ringel, C.M.: The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996), pp. 467鈥?80. In: CMS Conference Proceedings, 24, American Mathematical Society, Providence, RI (1998) - 作者单位：Yuya Mizuno (1)
1. Graduate School of Mathematics, Nagoya University, Frocho, Chikusaku, Nagoya, 464-8602, Japan - ISSN：1432-1823
文摘
We study support \(\tau \) -tilting modules over preprojective algebras of Dynkin type. We classify basic support \(\tau \) -tilting modules by giving a bijection with elements in the corresponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study \(g\) -matrices of support \(\tau \) -tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit description of \(g\) -matrices and prove that cones given by \(g\) -matrices coincide with chambers of the associated root systems.