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...
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| Format: | Article |
| Language: | English |
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Springer-Verlag
2016
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| Online Access: | http://hdl.handle.net/1721.1/104854 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826199830044606464 |
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| author | Walton, Chelsea Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Walton, Chelsea Etingof, Pavel I |
| author_sort | Walton, Chelsea |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T11:26:25Z |
| format | Article |
| id | mit-1721.1/104854 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T11:26:25Z |
| publishDate | 2016 |
| publisher | Springer-Verlag |
| record_format | dspace |
| spelling | mit-1721.1/1048542022-10-01T03:40:09Z Pointed Hopf Actions On Fields, I Walton, Chelsea Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel 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 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. National Science Foundation (U.S.) (DMS-1000173) National Science Foundation (U.S.) (DMS-1102548) National Science Foundation (U.S.) (DMS-1401207) 2016-10-19T18:01:45Z 2016-10-19T18:01:45Z 2015-08 2014-04 2016-08-18T15:41:30Z Article http://purl.org/eprint/type/JournalArticle 1083-4362 1531-586X http://hdl.handle.net/1721.1/104854 Etingof, Pavel, and Chelsea Walton. “Pointed Hopf Actions On Fields, I.” Transformation Groups vol. 20, no. 4, August 2015, pp. 985–1013. https://orcid.org/0000-0002-0710-1416 en http://dx.doi.org/10.1007/s00031-015-9328-7 Transformation Groups Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ Springer Science+Business Media New York application/pdf Springer-Verlag Springer US |
| spellingShingle | Walton, Chelsea Etingof, Pavel I Pointed Hopf Actions On Fields, I |
| title | Pointed Hopf Actions On Fields, I |
| title_full | Pointed Hopf Actions On Fields, I |
| title_fullStr | Pointed Hopf Actions On Fields, I |
| title_full_unstemmed | Pointed Hopf Actions On Fields, I |
| title_short | Pointed Hopf Actions On Fields, I |
| title_sort | pointed hopf actions on fields i |
| url | http://hdl.handle.net/1721.1/104854 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT waltonchelsea pointedhopfactionsonfieldsi AT etingofpaveli pointedhopfactionsonfieldsi |