AR-components for generalized Beilinson algebras

Worch, Julia

doi:10.1090/S0002-9939-2015-12621-4

AR-components for generalized Beilinson algebras
HTML articles powered by AMS MathViewer

by Julia Worch

Proc. Amer. Math. Soc. 143 (2015), 4271-4281

DOI: https://doi.org/10.1090/S0002-9939-2015-12621-4

Published electronically: March 31, 2015

PDF | Request permission

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 $\Gamma (n,r)$.

References

Ibrahim Assem, Daniel Simson, and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 1, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006. Techniques of representation theory. MR 2197389, DOI 10.1017/CBO9780511614309
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, DOI 10.1017/CBO9780511623608
A. A. Beĭlinson, Coherent sheaves on $\textbf {P}^{n}$ and problems in linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 68–69 (Russian). MR 509388
Jon F. Carlson, Eric M. Friedlander, and Julia Pevtsova, Modules of constant Jordan type, J. Reine Angew. Math. 614 (2008), 191–234. MR 2376286, DOI 10.1515/CRELLE.2008.006
Jon F. Carlson, Eric M. Friedlander, and Andrei Suslin, Modules for $\Bbb Z/p\times \Bbb Z/p$, Comment. Math. Helv. 86 (2011), no. 3, 609–657. MR 2803855, DOI 10.4171/CMH/236
Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107, DOI 10.1007/BFb0084003
R. Farnsteiner, Categories of modules given by varieties of $p$-nilpotent operators. Preprint: arXiv:1110.2706.
O. Kerner, Private communication, June 2013.
Claus Michael Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), no. 3, 235–255. MR 501169, DOI 10.1007/BF01214506
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
Daniel Simson and Andrzej Skowroński, Elements of the representation theory of associative algebras. Vol. 3, London Mathematical Society Student Texts, vol. 72, Cambridge University Press, Cambridge, 2007. Representation-infinite tilted algebras. MR 2382332
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, DOI 10.1112/jlms/jdt039
J. Worch, Module categories and Auslander-Reiten theory for generalized Beilinson algebras. http://macau.uni-kiel.de/receive/dissertationdiss00013419, 2013.