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arXiv · 2604.04813

Drinfeld-Xu bialgebroid 2-cocycles twist the antipode

Zoran Škoda
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Abstract

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.

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Zoran Škoda. 2026-04-06. Drinfeld-Xu bialgebroid 2-cocycles twist the antipode. https://arxiv.org/abs/2604.04813

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