Finite Dimensional Hopf Actions on Central Division Algebras
Let k be an algebraically closed field of characteristic zero. Let D be a division algebra of degree d over its center Z(D). Assume that k. Z(D). We show that a finite group G faithfully grades D if and only if G contains a normal abelian subgroup of index dividing d. We also prove that if a finite...
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| Format: | Article |
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Oxford University Press (OUP)
2018
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| Online Access: | http://hdl.handle.net/1721.1/115878 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826192809363767296 |
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| author | Cuadra-Diaz, Juan Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Cuadra-Diaz, Juan Etingof, Pavel I |
| author_sort | Cuadra-Diaz, Juan |
| collection | MIT |
| description | Let k be an algebraically closed field of characteristic zero. Let D be a division algebra of degree d over its center Z(D). Assume that k. Z(D). We show that a finite group G faithfully grades D if and only if G contains a normal abelian subgroup of index dividing d. We also prove that if a finite dimensional Hopf algebra coacts on D defining a Hopf-Galois extension, then its PI degree is at most d². Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles. |
| first_indexed | 2024-09-23T09:29:12Z |
| format | Article |
| id | mit-1721.1/115878 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T09:29:12Z |
| publishDate | 2018 |
| publisher | Oxford University Press (OUP) |
| record_format | dspace |
| spelling | mit-1721.1/1158782022-09-30T14:45:16Z Finite Dimensional Hopf Actions on Central Division Algebras Cuadra-Diaz, Juan Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Cuadra-Diaz, Juan Etingof, Pavel I Let k be an algebraically closed field of characteristic zero. Let D be a division algebra of degree d over its center Z(D). Assume that k. Z(D). We show that a finite group G faithfully grades D if and only if G contains a normal abelian subgroup of index dividing d. We also prove that if a finite dimensional Hopf algebra coacts on D defining a Hopf-Galois extension, then its PI degree is at most d². Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles. 2018-05-24T20:10:37Z 2018-05-24T20:10:37Z 2016-05 2015-08 2018-05-21T14:38:29Z Article http://purl.org/eprint/type/JournalArticle 1073-7928 1687-0247 http://hdl.handle.net/1721.1/115878 Cuadra, Juan and Pavel Etingof. “Finite Dimensional Hopf Actions on Central Division Algebras.” International Mathematics Research Notices 5, 1 (May 2016): 1562–1577 © 2016 The Author(s) https://orcid.org/0000-0002-0710-1416 http://dx.doi.org/10.1093/IMRN/RNW030 International Mathematics Research Notices Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press (OUP) arXiv |
| spellingShingle | Cuadra-Diaz, Juan Etingof, Pavel I Finite Dimensional Hopf Actions on Central Division Algebras |
| title | Finite Dimensional Hopf Actions on Central Division Algebras |
| title_full | Finite Dimensional Hopf Actions on Central Division Algebras |
| title_fullStr | Finite Dimensional Hopf Actions on Central Division Algebras |
| title_full_unstemmed | Finite Dimensional Hopf Actions on Central Division Algebras |
| title_short | Finite Dimensional Hopf Actions on Central Division Algebras |
| title_sort | finite dimensional hopf actions on central division algebras |
| url | http://hdl.handle.net/1721.1/115878 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT cuadradiazjuan finitedimensionalhopfactionsoncentraldivisionalgebras AT etingofpaveli finitedimensionalhopfactionsoncentraldivisionalgebras |