Drinfeld-Xu bialgebroid 2-cocycles twist the antipode
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914448933912576 |
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| author | Škoda, Zoran |
| author_facet | Škoda, Zoran |
| contents | Ping Xu generalized Drinfeld 2-cocycles from bialgebras to associative bialgebroids over noncommutative base algebras. Any counital Drinfeld--Xu 2-cocycle twists the base algebra of the bialgebroid and a comultiplication on the total algebra, obtaining a new, twisted bialgebroid. Antipodes for bialgebroids have been considered, but finding a general way to twist the antipode, which is straightforward in the Hopf algebra case, appeared somewhat elusive. In this article, we prove that if an invertible antipode $S$ for the original bialgebroid exists, and another expression $V_F$ depending on the 2-cocycle $F$ is invertible, then the expected conjugation formula $S_F(-) = V_F^{-1} S(-) V_F$ indeed produces an invertible antipode $S_F$ for the twisted bialgebroid. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04813 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Drinfeld-Xu bialgebroid 2-cocycles twist the antipode Škoda, Zoran Quantum Algebra Rings and Algebras Ping Xu generalized Drinfeld 2-cocycles from bialgebras to associative bialgebroids over noncommutative base algebras. Any counital Drinfeld--Xu 2-cocycle twists the base algebra of the bialgebroid and a comultiplication on the total algebra, obtaining a new, twisted bialgebroid. Antipodes for bialgebroids have been considered, but finding a general way to twist the antipode, which is straightforward in the Hopf algebra case, appeared somewhat elusive. In this article, we prove that if an invertible antipode $S$ for the original bialgebroid exists, and another expression $V_F$ depending on the 2-cocycle $F$ is invertible, then the expected conjugation formula $S_F(-) = V_F^{-1} S(-) V_F$ indeed produces an invertible antipode $S_F$ for the twisted bialgebroid. |
| title | Drinfeld-Xu bialgebroid 2-cocycles twist the antipode |
| topic | Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2604.04813 |