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Representations of constant socle rank for the Kronecker algebra

  • Daniel Bissinger EMAIL logo
Published/Copyright: September 11, 2019

Abstract

Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for p-elementary abelian groups Er of rank r over a field of characteristic p>0, we introduce the notions of modules with constant d-radical rank and modules with constant d-socle rank for the generalized Kronecker algebra š’¦r=k⁢Γr with r≄2 arrows and 1≤d≤r-1. We study subcategories given by modules with the equal d-radical property and the equal d-socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories in modā”š’¦r are of wild type. Then we use a natural functor š”‰:modā”š’¦r→mod⁔k⁢Er to transfer our results to mod⁔k⁢Er.

MSC 2010: 16G20; 16G60; 16G70

Communicated by Karl-Hermann Neeb


Award Identifier / Grant number: DFG priority program SPP 1388

Funding statement: Partly supported by the DFG priority program SPP 1388 ā€œDarstellungstheorieā€.

Acknowledgements

The results of this article are part of my doctoral thesis, which I have written at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for fruitful discussions, his continuous support and helpful comments on an earlier version of this paper. I also would like to thank the whole research team for the very pleasant working atmosphere and the encouragement throughout my studies. Furthermore, I thank Otto Kerner for answering my questions on hereditary algebras and giving helpful comments, and Claus Michael Ringel for sharing his insights on elementary modules for the Kronecker algebra. I would like to thank the anonymous referee for the detailed comments.

References

[1] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. I, London Math. Soc. Stud. Texts 72, Cambridge University, Cambridge, 2006. 10.1017/CBO9780511614309Search in Google Scholar

[2] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras. III, London Math. Soc. Stud. Texts 72, Cambridge University Press, Cambridge, 2007. 10.1017/CBO9780511619212Search in Google Scholar

[3] D. J. Benson, Representations of Elementary Abelian p-groups and Vector Bundles, Cambridge Tracts in Math. 208, Cambridge University, Cambridge, 2017. 10.1017/9781316795699Search in Google Scholar

[4] V. M. Bondarenko and I. V. Lytvynchuk, The representation type of elementary abelian p-groups with respect to the modules of constant Jordan type, Algebra Discrete Math. 14 (2012), no. 1, 29–36. Search in Google Scholar

[5] J. F. Carlson, E. M. Friedlander and J. Pevtsova, Modules of constant Jordan type, J. Reine Angew. Math. 614 (2008), 191–234. 10.1515/CRELLE.2008.006Search in Google Scholar

[6] J. F. Carlson, E. M. Friedlander and J. Pevtsova, Representations of elementary abelian p-groups and bundles on Grassmannians, Adv. Math. 229 (2012), no. 5, 2985–3051. 10.1016/j.aim.2011.11.004Search in Google Scholar

[7] B. Chen, Dimension vectors in regular components over wild Kronecker quivers, Bull. Sci. Math. 137 (2013), 730–745. 10.1016/j.bulsci.2013.04.002Search in Google Scholar

[8] P. Donovan and M. R. Freislich, The Representation Theory of Finite Graphs and Associated Algebras, Carleton Math. Lecture Notes 5, Carleton University, Ottawa, 1973. Search in Google Scholar

[9] R. Farnsteiner, Categories of modules given by varieties of p-nilpotent operators, preprint (2011), https://arxiv.org/abs/1110.2706. Search in Google Scholar

[10] R. Farnsteiner, Nilpotent operators, categories of modules, and auslander-reiten theory, Lectures notes (2012), http://www.math.uni-kiel.de/algebra/de/farnsteiner/material/Shanghai-2012-Lectures.pdf. Search in Google Scholar

[11] P. Gabriel, Indecomposable representations. II, Symposia Mathematica Vol. XI (Rome 1971), Academic Press, London (1973), 81–104. Search in Google Scholar

[12] V. G. Kac, Root systems, representations of quivers and invariant theory, Invariant Theory (Montecatini 1982), Lecture Notes in Math. 996, Springer, Berlin (1983), 74–108. 10.1007/BFb0063236Search in Google Scholar

[13] O. Kerner, Representations of wild quivers, Representation Theory of Algebras and Related Topics (Mexico City 1994), CMS Conf. Proc. 19, American Mathematical Society, Providence (1996), 65–107. Search in Google Scholar

[14] O. Kerner, More representations of wild quivers, Expository Lectures on Representation Theory, Contemp. Math. 607, American Mathematical Society, Providence (2014), 35–55. 10.1090/conm/607/12087Search in Google Scholar

[15] O. Kerner and F. Lukas, Regular modules over wild hereditary algebras, Representations of Finite-dimensional Algebras (Tsukuba 1990), CMS Conf. Proc. 11, American Mathematical Society, Providence (1991), 191–208. Search in Google Scholar

[16] O. Kerner and F. Lukas, Elementary modules, Math. Z. 223 (1996), no. 3, 421–434. 10.1007/PL00004567Search in Google Scholar

[17] C. M. Ringel, Representations of K-species and bimodules, J. Algebra 41 (1976), no. 2, 269–302. 10.1016/0021-8693(76)90184-8Search in Google Scholar

[18] C. M. Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), no. 3, 235–255. 10.1007/BF01214506Search in Google Scholar

[19] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, Berlin, 1984. 10.1007/BFb0072870Search in Google Scholar

[20] L. Unger, The concealed algebras of the minimal wild, hereditary algebras, Bayreuth. Math. Schr. (1990), no. 31, 145–154. Search in Google Scholar

[21] J. Worch, Categories of modules for elementary abelian p-groups and generalized Beilinson algebras, J. Lond. Math. Soc. (2) 88 (2013), no. 3, 649–668. 10.1112/jlms/jdt039Search in Google Scholar

[22] J. Worch, Module categories and Auslander–Reiten theory for generalized Beilinson algebras, PhD thesis, Christian-Albrechts-UniversitƤt zu Kiel, 2013. Search in Google Scholar

[23] J. Worch, AR-components for generalized Beilinson algebras, Proc. Amer. Math. Soc. 143 (2015), no. 10, 4271–4281. 10.1090/S0002-9939-2015-12621-4Search in Google Scholar

Received: 2018-06-14
Revised: 2019-07-11
Published Online: 2019-09-11
Published in Print: 2020-01-01

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