Higher representation infinite algebras from McKay quivers of metacyclic groups. (English) Zbl 1471.16022
Summary: For each prime number \(s\) we introduce examples of \(s-1\)- and \(s\)-representation infinite algebras in the sense of Herschend, Iyama and Oppermann
[M. Herschend et al., Adv. Math. 252, 292–342 (2014; Zbl 1339.16020)], which arise from skew group algebras of some metacyclic groups embedded in \(\mathrm{SL}(s,\mathbb{C})\) and \(\mathrm{SL}(s+1,\mathbb{C)}\). For this purpose, we give a description of the McKay quiver with a superpotential of such groups. Moreover, we show that for \(s=2\) our examples correspond to the classical tame hereditary algebras of type \(\tilde{D}\).
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16S35 | Twisted and skew group rings, crossed products |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
Keywords:
\(n\)-representation infinite algebra; higher preprojective algebra; skew group algebra; McKay quiver; metacyclic groupCitations:
Zbl 1339.16020References:
| [1] | Amiot, C.; Iyama, O.; Reiten, I., Stable categories of Cohen-Macaulay modules and cluster categories, Amer. J. Math, 137, 3, 813-857 (2015) · Zbl 1450.18003 · doi:10.1353/ajm.2015.0019 |
| [2] | Benson, D., Cambridge Studies in Advanced Mathematics, 30p, Representations and Cohomology I (1991), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0731.20001 |
| [3] | Bocklandt, R.; Schedler, T.; Wemyss, M., Superpotentials and higher order derivations, J. Pure Appl. Algebra, 214, 9, 1501-1522 (2010) · Zbl 1219.16016 · doi:10.1016/j.jpaa.2009.07.013 |
| [4] | Bondal, A. I.; Kapranov, M. M., Representable functors, Serre functors, and reconstructions, Math. Ussr. Izv., 35, 3, 519-541 (1990) · Zbl 0703.14011 · doi:10.1070/IM1990v035n03ABEH000716 |
| [5] | Curtis, C. W.; Reiner, I., Representation Theory of Finite Groups and Associative Algebras. (1962), New York, NY: Wiley & Sons-Interscience, New York, NY · Zbl 0131.25601 |
| [6] | Donovan, P.; Freislich, M. R., Carleton Math. Lecture Notes, 5, The representation theory of finite graphs and associated algebras (1973), Ottawa, ON: Carleton University, Ottawa, ON · Zbl 0304.08006 |
| [7] | Gel’Fand, I. M.; Ponomarev, V. A., Model algebras and representations of graphs, Funct. Anal. Appl, 13, 157 (1979) · Zbl 0437.16020 · doi:10.1007/BF01077482 |
| [8] | Guo, J. Y., On McKay quivers and covering spaces, Sci. Sin. Math, 41, 5, 393-402 (2011) · Zbl 1488.16041 · doi:10.1360/012009-219 |
| [9] | Herschend, M.; Iyama, O.; Minamoto, H.; Oppermann, S., Representation theory of Geigle-Lenzing complete intersections, arXiv:1409.0668 · Zbl 1530.16001 |
| [10] | Herschend, M.; Iyama, O.; Oppermann, S., n-representation infinite algebras, Adv. Math, 252, 292-342 (2014) · Zbl 1339.16020 · doi:10.1016/j.aim.2013.09.023 |
| [11] | Humphreys, J. E., Cambridge Studies in Advanced Mathematics, 29, Reflection groups and Coxeter groups (1990), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0725.20028 |
| [12] | Ito, Y., Crepant resolution of trihedral singularities, Proc. Japan Acad. Ser. A Math. Sci., 70, 5, 131-136 (1994) · Zbl 0831.14006 · doi:10.3792/pjaa.70.131 |
| [13] | Ito, Y.; Reid, M.; Andreatta, M.; Peternell, T., Higher Dimensional Complex Varieties (Trento, Jun 1994), The McKay correspondence for finite subgroups of SL(3, C), 221-240 (1996), Berlin, Germany: de Gruyter, pp, Berlin, Germany · Zbl 0894.14024 |
| [14] | Karmazyn, J., Superpotentials, Calabi-Yau algebras, and PBW deformations, J. Algebra, 413, 100-134 (2014) · Zbl 1334.16011 · doi:10.1016/j.jalgebra.2014.05.007 |
| [15] | Keller, B., Deformed Calabi-Yau completions. With an appendix by Michel van den Bergh, J. Reine Angew. Math, 654, 125-180 (2011) · Zbl 1220.18012 |
| [16] | Leng, R., The McKay correspondence and orbifold Riemann-Roch (2002) |
| [17] | Mckay, J., Graphs, singularities, and finite groups, Proc. Symp. Pure Math., 37, 183-186 (1980) · Zbl 0451.05026 |
| [18] | Minamoto, H.; Mori, I., Structures of as-regular algebras, Adv. Math., 226, 5, 4061-4095 (2011) · Zbl 1228.16023 · doi:10.1016/j.aim.2010.11.004 |
| [19] | Nazarova, L. A., Representations of quivers of infinite type, Math. Ussr. Izv., 7, 4, 749-792 (1973) · Zbl 0343.15004 · doi:10.1070/IM1973v007n04ABEH001975 |
| [20] | Nolla De Celis, A., Dihedral groups and G-Hilbert schemes (2008), Coventry, UK |
| [21] | Reiten, I.; Van Den Bergh, M., Two-dimensional tame and maximal orders of finite representation type, Mem. Amer. Math. Soc., 80, 408 (1989) · Zbl 0677.16002 · doi:10.1090/memo/0408 |
| [22] | Ringel, C. M. (1998). The preprojective algebra of a quiver, Algebras and modules, II (Geiranger, 1996). Presented at CMS Conf. Proc., Vol. 24, Amer. Math. Soc., Providence, RI, pp. 467-480. · Zbl 0928.16012 |
| [23] | Thibault, L. P., Preprojective algebra structure on skew-group algebras, arXiv:1603.04324 · Zbl 1439.16010 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
