Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-04T19:01:09.567Z Has data issue: false hasContentIssue false
  • English
  • Français

DISTRIBUTIVE LATTICES OF TILTING MODULES AND SUPPORT τ-TILTING MODULES OVER PATH ALGEBRAS

Part of: Representation theory of rings and algebras Algebraic combinatorics

Published online by Cambridge University Press:  10 June 2016

YICHAO YANG
YICHAO YANG*
Affiliation:
Département de mathématiques, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1 E-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$.

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 05E10: Combinatorial aspects of representation theory
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 503 - 511
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Assem, I., Brüstle, T. and Schiffler, R., Cluster-tilted algebras and slices, J. Algebra 319 (8) (2008), 3464–3479.Google Scholar
2. Adachi, T., Iyama, O. and Reiten, I., τ-tilting theory, Compos. Math. 150 (3) (2014), 415–452.Google Scholar
3. Assem, I., Simson, D. and Skowroński, A., Elements of the representation theory of associative algebras. Vol. 1. Techniques of representation theory, London Mathematical Society Student Texts, vol. 65 (Cambridge University Press, Cambridge, 2006).Google Scholar
4. Brenner, S. and Butler, M. C. R., Generalization of the Bernstein-Gelfand-Ponomarev reflection functors, Lecture Notes in Math., vol. 839 (Springer-Verlag, Berlin, 1980), 103–169.Google Scholar
5. Happel, D. and Unger, L., On a partial order of tilting modules, Algebr. Represent. Theory 8 (2) (2005), 147–156.Google Scholar
6. Happel, D. and Vossieck, D., Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (2–3) (1983), 221–243.Google Scholar
7. Iyama, O., Reiten, I., Thomas, H. and Todorov, G., Lattice structure of torsion classes for path algebras, Bull. Lond. Math. Soc. 47 (4) (2015), 639–650.Google Scholar
8. Ingalls, C. and Thomas, H., Noncrossing partitions and representations of quivers, Compos. Math. 145 (6) (2009), 1533–1562.Google Scholar
9. Kase, R., Distributive lattices and the poset of pre-projective tilting modules, J. Algebra 415 (1) (2014), 264–289.Google Scholar
10. Kerner, O. and Takane, M., Mono orbits, epi orbits and elementary vertices of representation infinite quivers, Comm. Algebra 25 (1) (1997), 51–77.Google Scholar
11. Liu, S., Shapes of connected components of the Auslander-Reiten quivers of artin algebras, in Representation theory of algebras and related topics (Mexico City, 1994); Canad. Math. Soc. Conf. Proc., vol. 19 (1995), 109–137.Google Scholar
12. Liu, S., Another characterization of tilted algebras, Arch. Math. 104 (2) (2015), 111–123.Google Scholar
13. Li, F. and Yang, Y. C., A note on section and slice for a hereditary algebra, Int. J. Appl. Math. Stat. 52 (9) (2014), 112–119.Google Scholar
14. Ringel, C. M., Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099 (Springer-Verlag, Berlin, 1984).Google Scholar
15. Ringel, C. M., Lattice structure of torsion classes for hereditary artin algebras, arXiv:1402.1260.Google Scholar
16. Riedtmann, C. and Schofield, A., On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1) (1991), 70–78.Google Scholar
17. Zhang, P., Separating tilting modules, Chinese Sci. Bull. 37 (12) (1992), 975–978.Google Scholar