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AR-components for generalized Beilinson algebras


Author: Julia Worch
Journal: Proc. Amer. Math. Soc. 143 (2015), 4271-4281
MSC (2010): Primary 16G20, 16G70; Secondary 16S90, 16S37
DOI: https://doi.org/10.1090/S0002-9939-2015-12621-4
Published electronically: March 31, 2015
MathSciNet review: 3373926
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Abstract: We show that the generalized $ W$-modules defined in 2013 determine $ \mathbb{Z}A_{\infty }$-components in the Auslander-Reiten quiver $ \Gamma (n,r)$ of the generalized Beilinson algebra $ B(n,r)$, $ n \geq 3$. These components entirely consist of modules with the constant Jordan type property. We arrive at this result by interpreting $ B(n,r)$ as an iterated one-point extension of the $ r$-Kronecker algebra $ \mathcal {K}_r$, which enables us to generalize findings concerning the Auslander-Reiten quiver $ \Gamma (\mathcal {K}_r)$ presented in 2013 to $ \Gamma (n,r)$.


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  • [1] Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, 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. MR 2197389 (2006j:16020)
  • [2] Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422 (96c:16015)
  • [3] A. A. Beĭlinson, Coherent sheaves on $ {\bf P}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68-69 (Russian). MR 509388 (80c:14010b)
  • [4] Jon F. Carlson, Eric M. Friedlander, and Julia Pevtsova, Modules of constant Jordan type, J. Reine Angew. Math. 614 (2008), 191-234. MR 2376286 (2008j:20135), https://doi.org/10.1515/CRELLE.2008.006
  • [5] Jon F. Carlson, Eric M. Friedlander, and Andrei Suslin, Modules for $ \mathbb{Z}/p\times \mathbb{Z}/p$, Comment. Math. Helv. 86 (2011), no. 3, 609-657. MR 2803855 (2012d:20017), https://doi.org/10.4171/CMH/236
  • [6] Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107 (91c:20016)
  • [7] R. Farnsteiner, Categories of modules given by varieties of $ p$-nilpotent operators.
    Preprint: arXiv:1110.2706.
  • [8] O. Kerner,
    Private communication, June 2013.
  • [9] Claus Michael Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), no. 3, 235-255. MR 501169 (80c:16017), https://doi.org/10.1007/BF01214506
  • [10] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
  • [11] Daniel Simson and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 3, Representation-infinite tilted algebras. London Mathematical Society Student Texts, vol. 72, Cambridge University Press, Cambridge, 2007. MR 2382332 (2008m:16001)
  • [12] Julia Worch, Categories of modules for elementary abelian $ p$-groups and generalized Beilinson algebras, J. Lond. Math. Soc. (2) 88 (2013), no. 3, 649-668. MR 3145125, https://doi.org/10.1112/jlms/jdt039
  • [13] J. Worch, Module categories and Auslander-Reiten theory for generalized Beilinson algebras.
    http://macau.uni-kiel.de/receive/dissertationdiss00013419, 2013.

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Additional Information

Julia Worch
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany
Email: jworch@gmx.net

DOI: https://doi.org/10.1090/S0002-9939-2015-12621-4
Received by editor(s): January 23, 2014
Received by editor(s) in revised form: June 22, 2014
Published electronically: March 31, 2015
Additional Notes: The author’s research was partly supported by the D.F.G. priority program SPP 1388 “Darstellungstheorie”
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2015 American Mathematical Society

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