Finite dimensional Hopf actions on deformation quantizations

Etingof, Pavel; Walton, Chelsea

doi:10.1090/proc/13356

Finite dimensional Hopf actions on deformation quantizations
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by Pavel Etingof and Chelsea Walton PDF

Proc. Amer. Math. Soc. 145 (2017), 1917-1925 Request permission

Abstract:

We study when a finite dimensional Hopf action on a quantum formal deformation $A$ of a commutative domain $A_0$ (i.e., a deformation quantization) must factor through a group algebra. In particular, we show that this occurs when the Poisson center of the fraction field of $A_0$ is trivial.

References

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
R. Bezrukavnikov and D. Kaledin, Fedosov quantization in algebraic context, Mosc. Math. J. 4 (2004), no. 3, 559–592, 782 (English, with English and Russian summaries). MR 2119140, DOI 10.17323/1609-4514-2004-4-3-559-592
Jeffrey Bergen, Miriam Cohen, and Davida Fischman, Irreducible actions and faithful actions of Hopf algebras, Israel J. Math. 72 (1990), no. 1-2, 5–18. Hopf algebras. MR 1098978, DOI 10.1007/BF02764609
K. Chan, C. Walton, Y. H. Wang, and J. J. Zhang, Hopf actions on filtered regular algebras, J. Algebra 397 (2014), 68–90. MR 3119216, DOI 10.1016/j.jalgebra.2013.09.002
Juan Cuadra, Pavel Etingof, and Chelsea Walton, Semisimple Hopf actions on Weyl algebras, Adv. Math. 282 (2015), 47–55. MR 3374522, DOI 10.1016/j.aim.2015.05.014
J. Cuadra, P. Etingof, and C. Walton, Finite dimensional Hopf actions on Weyl algebras, arXiv:1509.01165.
P. Etingof, D. Goswami, A. Mandal, and C. Walton, Hopf coactions on commutative algebras generated by a quadratically independent comodule, arXiv:1507.08486.
Pavel Etingof and Chelsea Walton, Semisimple Hopf actions on commutative domains, Adv. Math. 251 (2014), 47–61. MR 3130334, DOI 10.1016/j.aim.2013.10.008
Pavel Etingof and Chelsea Walton, Pointed Hopf actions on fields, I, Transform. Groups 20 (2015), no. 4, 985–1013. MR 3416436, DOI 10.1007/s00031-015-9328-7
Serge Skryabin and Freddy Van Oystaeyen, The Goldie Theorem for $H$-semiprime algebras, J. Algebra 305 (2006), no. 1, 292–320. MR 2264132, DOI 10.1016/j.jalgebra.2006.06.030