Group coactions on two-dimensional Artin-Schelter regular algebras

Crawford, Simon

doi:10.1090/proc/16844

Group coactions on two-dimensional Artin-Schelter regular algebras
HTML articles powered by AMS MathViewer

by Simon Crawford;

Proc. Amer. Math. Soc. 152 (2024), 4551-4567

DOI: https://doi.org/10.1090/proc/16844

Published electronically: September 4, 2024

HTML | PDF

Abstract:

We describe all possible coactions of finite groups (equivalently, all group gradings) on two-dimensional Artin-Schelter regular algebras. We give necessary and sufficient conditions for the associated Auslander map to be an isomorphism, and determine precisely when the invariant ring for the coaction is Artin-Schelter regular. The proofs of our results are combinatorial and exploit the structure of the McKay quiver associated to the coaction.

References

Michael Artin and William F. Schelter, Graded algebras of global dimension $3$, Adv. in Math. 66 (1987), no. 2, 171–216. MR 917738, DOI 10.1016/0001-8708(87)90034-X
M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33–85. MR 1086882
Maurice Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293 (1986), no. 2, 511–531. MR 816307, DOI 10.1090/S0002-9947-1986-0816307-7
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
Yanhong Bao, Jiwei He, and James J. Zhang, Pertinency of Hopf actions and quotient categories of Cohen-Macaulay algebras, J. Noncommut. Geom. 13 (2019), no. 2, 667–710. MR 3988759, DOI 10.4171/JNCG/336
Claude Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782. MR 72877, DOI 10.2307/2372597
K. Chan, E. Kirkman, C. Walton, and J. J. Zhang, McKay correspondence for semisimple Hopf actions on regular graded algebras, I, J. Algebra 508 (2018), 512–538. MR 3810305, DOI 10.1016/j.jalgebra.2018.05.008
Kenneth Chan, Ellen Kirkman, Chelsea Walton, and James J. Zhang, McKay correspondence for semisimple Hopf actions on regular graded algebras. II, J. Noncommut. Geom. 13 (2019), no. 1, 87–114. MR 3941474, DOI 10.4171/JNCG/305
J. Chen, E. Kirkman, and J. J. Zhang, Rigidity of down-up algebras with respect to finite group coactions, J. Pure Appl. Algebra 221 (2017), no. 12, 3089–3103. MR 3666738, DOI 10.1016/j.jpaa.2017.02.015
Simon Crawford, Actions of small groups on two-dimensional Artin-Schelter regular algebras, J. Algebra 560 (2020), 609–666. MR 4110218, DOI 10.1016/j.jalgebra.2020.05.005
Simon Crawford, Superpotentials and quiver algebras for semisimple Hopf actions, Algebr. Represent. Theory 26 (2023), no. 6, 2709–2752. MR 4681330, DOI 10.1007/s10468-022-10165-y
Luigi Ferraro, Ellen Kirkman, W. Frank Moore, and Robert Won, Three infinite families of reflection Hopf algebras, J. Pure Appl. Algebra 224 (2020), no. 8, 106315, 34. MR 4074572, DOI 10.1016/j.jpaa.2020.106315
Peter Jørgensen and James J. Zhang, Gourmet’s guide to Gorensteinness, Adv. Math. 151 (2000), no. 2, 313–345. MR 1758250, DOI 10.1006/aima.1999.1897
Ellen E. Kirkman, Invariant theory of Artin-Schelter regular algebras: a survey, Recent developments in representation theory, Contemp. Math., vol. 673, Amer. Math. Soc., Providence, RI, 2016, pp. 25–50. MR 3546709, DOI 10.1090/conm/673/13489
E. Kirkman, J. Kuzmanovich, and J. J. Zhang, Gorenstein subrings of invariants under Hopf algebra actions, J. Algebra 322 (2009), no. 10, 3640–3669. MR 2568355, DOI 10.1016/j.jalgebra.2009.08.018
Günter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Revised edition, Graduate Studies in Mathematics, vol. 22, American Mathematical Society, Providence, RI, 2000. MR 1721834, DOI 10.1090/gsm/022
Susan Montgomery, Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, vol. 82, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1993. MR 1243637, DOI 10.1090/cbms/082
D. S. Passman and Declan Quinn, Burnside’s theorem for Hopf algebras, Proc. Amer. Math. Soc. 123 (1995), no. 2, 327–333. MR 1215204, DOI 10.1090/S0002-9939-1995-1215204-6
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
D. R. Stephenson, Artin-Schelter regular algebras of global dimension three, J. Algebra 183 (1996), no. 1, 55–73. MR 1397387, DOI 10.1006/jabr.1996.0207
Darin R. Stephenson, Algebras associated to elliptic curves, Trans. Amer. Math. Soc. 349 (1997), no. 6, 2317–2340. MR 1390046, DOI 10.1090/S0002-9947-97-01769-8
Michael Wemyss, The $\textrm {GL}(2,\Bbb C)$ McKay correspondence, Math. Ann. 350 (2011), no. 3, 631–659. MR 2805639, DOI 10.1007/s00208-010-0572-9
Ruipeng Zhu, Auslander theorem for PI Artin-Schelter regular algebras, Proc. Amer. Math. Soc. 151 (2023), no. 9, 3705–3719. MR 4607617, DOI 10.1090/proc/16424