Pointed Hopf Actions On Fields, I
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31_2015_Article_9328.pdf
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Author(s)
Walton, Chelsea
Etingof, Pavel I
Date Issued
August 2015
Journal
Transformation Groups
Publisher
Springer-Verlag
Citation
Etingof, Pavel, and Chelsea Walton. “Pointed Hopf Actions On Fields, I.” Transformation Groups vol. 20, no. 4, August 2015, pp. 985–1013.
Version
Author's final manuscript
Abstract
Actions of semisimple Hopf algebras H over an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors in [18]. The answer turns out to be very simple–if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type.
The work begins by reducing to the case where H acts inner faithfully on a field; such a Hopf algebra is referred to as Galois-theoretical. We present examples of such Hopf algebras, which include the Taft algebras, uq(sl₂), and some Drinfeld twists of other small quantum groups. We also give many examples of finite dimensional Hopf algebras which are not Galois-theoretical. Classification results on finite dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type will be provided in the sequel, Part II, of this study.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00031-015-9328-7
