Czechoslovak Mathematical Journal, first online, pp. 1-19


The duality of Auslander-Reiten quiver of path algebras

Bo Hou, Shilin Yang

Received November 28, 2017.   Published online February 18, 2019.

Abstract:  Let $Q$ be a finite union of Dynkin quivers, $G\subseteq{\rm Aut}(\Bbbk{Q})$ a finite abelian group, $\widehat{Q}$ the generalized McKay quiver of $(Q, G)$ and $\Gamma_Q$ the Auslander-Reiten quiver of $\Bbbk Q$. Then $G$ acts functorially on the quiver $\Gamma_Q$. We show that the Auslander-Reiten quiver of $\Bbbk\widehat{Q}$ coincides with the generalized McKay quiver of $(\Gamma_Q, G)$.
Keywords:  Auslander-Reiten quiver; generalized McKay quiver; duality
Classification MSC:  16G10, 16G20, 16G70
DOI:  10.21136/CMJ.2019.0541-17

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References:
[1] I. Assem, D. Simson, A. Skowroński: Elements of the Representation Theory of Associative Algebras. Vol. 1. Techniques of Representation Theory. London Mathematical Society Student Texts 65. Cambridge University Press, Cambridge (2006). DOI 10.1017/CBO9780511614309 | MR 2197389 | Zbl 1092.16001
[2] M. Auslander, I. Reiten, S. O. Smalø: Representation Theory of Artin Algebras. Cambridge Studies in Advanced Mathematics 36. Cambridge University Press, Cambridge (1995). DOI 10.1017/CBO9780511623608 | MR 1314422 | Zbl 0834.16001
[3] L. Demonet: Skew group algebras of path algebras and preprojective algebras. J. Algebra 323 (2010), 1052-1059. DOI 10.1016/j.jalgebra.2009.11.034 | MR 2578593 | Zbl 1210.16017
[4] B. Deng, J. Du: Frobenius morphisms and representations of algebras. Trans. Am. Math. Soc. 358 (2006), 3591-3622. DOI 10.1090/S0002-9947-06-03812-8 | MR 2218990 | Zbl 1095.16007
[5] B. Deng, J. Du, B. Parshall, J. Wang: Finite Dimensional Algebras and Quantum Groups. Mathematical Surveys and Monographs 150. American Mathematical Society, Providence (2008). DOI 10.1090/surv/150 | MR 2457938 | Zbl 1154.17003
[6] P. Gabriel, A. V. Roĭter: Algebra VIII. Representations of Finite-Dimensional Algebras (A. I. Kostrikin, et al.). Encyclopaedia of Mathematical Sciences 73. Springer, Berlin (1992). MR 1239447 | Zbl 0839.16001
[7] J. Guo: On the McKay quivers and $m$-Cartan matrices. Sci. China, Ser. A 52 (2009), 511-516. DOI 10.1007/s11425-008-0176-y | MR 2491769 | Zbl 1181.16014
[8] B. Hou, S. Yang: Skew group algebras of deformed preprojective algebras. J. Algebra 332 (2011), 209-228. DOI 10.1016/j.jalgebra.2011.02.007 | MR 2774685 | Zbl 1252.16010
[9] B. Hou, S. Yang: Generalized McKay quivers, root system and Kac-Moody algebras. J. Korean Math. Soc. 52 (2015), 239-268. DOI 10.4134/JKMS.2015.52.2.239 | MR 3318368 | Zbl 1335.16011
[10] A. Hubery: Representations of Quiver Respecting a Quiver Automorphism and a Theorem of Kac. Ph.D. Thesis, University of Leeds, Leeds (2002).
[11] A. Hubery: Quiver representations respecting a quiver automorphism: a generalization of a theorem of Kac. J. Lond. Math. Soc., II. Ser. 69 (2004), 79-96. DOI 10.1112/S0024610703004988 | MR 2025328 | Zbl 1062.16021
[12] V. G. Kac: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990). DOI 10.1017/CBO9780511626234 | MR 1104219 | Zbl 0716.17022
[13] G. X. Liu: Classification of Finite Dimensional Basic Hopf Algebras and Related Topics. Dissertation for the Doctoral Degree, Zhejiang University, Hangzhou (2005).
[14] J. McKay: Graphs, singularities, and finite groups. The Santa Cruz Conference on Finite Groups, Proc. Sympos. Pure Math. 37 American Mathematical Society, Providence (1980), 183-186. DOI 10.1090/pspum/037 | MR 0604577 | Zbl 0451.05026
[15] I. Reiten, C. Riedtmann: Skew group algebras in the representation theory of Artin algebras. J. Algebra 92 (1985), 224-282. DOI 10.1016/0021-8693(85)90156-5 | MR 0772481 | Zbl 0549.16017
[16] M. Zhang: The dual quiver of the Auslander-Reiten quiver of path algebras. Algebr. Represent. Theory 15 (2012), 203-210. DOI 10.1007/s10468-010-9237-3 | MR 2892506 | Zbl 1252.16015
[17] M. Zhang, F. Li: Representations of skew group algebras induced from isomorphically invariant modules over path algebras. J. Algebra 321 (2009), 567-581. DOI 10.1016/j.jalgebra.2008.09.035 | MR 2483282 | Zbl 1207.16015

Affiliations:   Bo Hou, School of Mathematics and Statistics, Henan University, Ming Lun Street, Kaifeng, Henan, China, e-mail: bohou1981@163.com; Shilin Yang (corresponding author), College of Applied Sciences, Beijing University of Technology, 100 Ping Le Yuan, Chaoyang District, Beijing 100124, China, e-mail: slyang@bjut.edu.cn


 
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