Finite dimensional Hopf actions on Weyl algebras
We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors,...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Published: |
Elsevier BV
2018
|
| Online Access: | http://hdl.handle.net/1721.1/119625 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1826212190953144320 |
|---|---|
| author | Cuadra, Juan Walton, Chelsea Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Cuadra, Juan Walton, Chelsea Etingof, Pavel I |
| author_sort | Cuadra, Juan |
| collection | MIT |
| description | We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operators. This gives an affirmative answer to a question posed by the last two authors. Keywords: Hopf algebra action; Weyl algebra; Algebra of differential operators; Reduction modulo prime powers |
| first_indexed | 2024-09-23T15:17:47Z |
| format | Article |
| id | mit-1721.1/119625 |
| institution | Massachusetts Institute of Technology |
| last_indexed | 2024-09-23T15:17:47Z |
| publishDate | 2018 |
| publisher | Elsevier BV |
| record_format | dspace |
| spelling | mit-1721.1/1196252022-09-29T14:01:59Z Finite dimensional Hopf actions on Weyl algebras Cuadra, Juan Walton, Chelsea Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I We prove that any action of a finite dimensional Hopf algebra H on a Weyl algebra A over an algebraically closed field of characteristic zero factors through a group action. In other words, Weyl algebras do not admit genuine finite quantum symmetries. This improves a previous result by the authors, where the statement was established for semisimple H. The proof relies on a refinement of the method previously used: namely, considering reductions of the action of H on A modulo prime powers rather than primes. We also show that the result holds, more generally, for algebras of differential operators. This gives an affirmative answer to a question posed by the last two authors. Keywords: Hopf algebra action; Weyl algebra; Algebra of differential operators; Reduction modulo prime powers National Science Foundation (U.S.) (Grant DMS-1000113) National Science Foundation (U.S.) (Grant DMS-1502244) National Science Foundation (U.S.) (Grant DMS-1550306) 2018-12-12T20:54:05Z 2018-12-12T20:54:05Z 2016-07 2016-04 2018-12-04T13:54:57Z Article http://purl.org/eprint/type/JournalArticle 0001-8708 1090-2082 http://hdl.handle.net/1721.1/119625 Cuadra, Juan et al. “Finite Dimensional Hopf Actions on Weyl Algebras.” Advances in Mathematics 302 (October 2016): 25–39 © 2016 Elsevier Inc. https://orcid.org/0000-0002-0710-1416 http://dx.doi.org/10.1016/J.AIM.2016.07.009 Advances in Mathematics Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier BV arXiv |
| spellingShingle | Cuadra, Juan Walton, Chelsea Etingof, Pavel I Finite dimensional Hopf actions on Weyl algebras |
| title | Finite dimensional Hopf actions on Weyl algebras |
| title_full | Finite dimensional Hopf actions on Weyl algebras |
| title_fullStr | Finite dimensional Hopf actions on Weyl algebras |
| title_full_unstemmed | Finite dimensional Hopf actions on Weyl algebras |
| title_short | Finite dimensional Hopf actions on Weyl algebras |
| title_sort | finite dimensional hopf actions on weyl algebras |
| url | http://hdl.handle.net/1721.1/119625 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT cuadrajuan finitedimensionalhopfactionsonweylalgebras AT waltonchelsea finitedimensionalhopfactionsonweylalgebras AT etingofpaveli finitedimensionalhopfactionsonweylalgebras |