On faithfulness of the lifting for Hopf algebras and fusion categories
We use a version of Haboush’s theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic p to characteristic zero, showing that, moreover, any isomorphism between...
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| Format: | Article |
| Language: | English |
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Mathematical Sciences Publishers
2019
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| Online Access: | https://hdl.handle.net/1721.1/122974 |
| _version_ | 1826189011865042944 |
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| author | Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I |
| author_sort | Etingof, Pavel I |
| collection | MIT |
| description | We use a version of Haboush’s theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic p to characteristic zero, showing that, moreover, any isomorphism between such structures can be reduced modulo p. This fills a gap in our earlier work. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer–Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from our earlier work), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo p. Keywords: lifting; Hopf algebra; tensor category; separable |
| first_indexed | 2024-09-23T08:08:27Z |
| format | Article |
| id | mit-1721.1/122974 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T08:08:27Z |
| publishDate | 2019 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | mit-1721.1/1229742022-09-23T11:09:57Z On faithfulness of the lifting for Hopf algebras and fusion categories Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Algebra and Number Theory We use a version of Haboush’s theorem over complete local Noetherian rings to prove faithfulness of the lifting for semisimple cosemisimple Hopf algebras and separable (braided, symmetric) fusion categories from characteristic p to characteristic zero, showing that, moreover, any isomorphism between such structures can be reduced modulo p. This fills a gap in our earlier work. We also show that lifting of semisimple cosemisimple Hopf algebras is a fully faithful functor, and prove that lifting induces an isomorphism on Picard and Brauer–Picard groups. Finally, we show that a subcategory or quotient category of a separable multifusion category is separable (resolving an open question from our earlier work), and use this to show that certain classes of tensor functors between lifts of separable categories to characteristic zero can be reduced modulo p. Keywords: lifting; Hopf algebra; tensor category; separable National Science Foundation (U.S.) (Grant DMS-1502244) 2019-11-20T14:20:08Z 2019-11-20T14:20:08Z 2018-06 2017-04 2019-11-12T17:25:32Z Article http://purl.org/eprint/type/JournalArticle 1944-7833 1937-0652 https://hdl.handle.net/1721.1/122974 Etingof, Pavel. "On faithfulness of the lifting for Hopf algebras and fusion categories." Algebra & Number Theory 12, 3 (June 2018): 551–569 © 2018 Mathematical Sciences Publishers en http://dx.doi.org/10.2140/ant.2018.12.551 Algebra & Number Theory Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Mathematical Sciences Publishers arXiv |
| spellingShingle | Algebra and Number Theory Etingof, Pavel I On faithfulness of the lifting for Hopf algebras and fusion categories |
| title | On faithfulness of the lifting for Hopf algebras and fusion categories |
| title_full | On faithfulness of the lifting for Hopf algebras and fusion categories |
| title_fullStr | On faithfulness of the lifting for Hopf algebras and fusion categories |
| title_full_unstemmed | On faithfulness of the lifting for Hopf algebras and fusion categories |
| title_short | On faithfulness of the lifting for Hopf algebras and fusion categories |
| title_sort | on faithfulness of the lifting for hopf algebras and fusion categories |
| topic | Algebra and Number Theory |
| url | https://hdl.handle.net/1721.1/122974 |
| work_keys_str_mv | AT etingofpaveli onfaithfulnessoftheliftingforhopfalgebrasandfusioncategories |