Pointed Hopf Actions On Fields, I
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 contri...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Springer-Verlag
2016
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| Online Access: | http://hdl.handle.net/1721.1/104854 https://orcid.org/0000-0002-0710-1416 |
| Summary: | 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. |
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