Drinfeld-Xu bialgebroid 2-cocycles twist the antipode

TL;DR

This paper extends the theory of Drinfeld-Xu 2-cocycles to associative bialgebroids, demonstrating how to explicitly twist the antipode when an invertible antipode exists, thus generalizing the Hopf algebra case.

Contribution

It provides a concrete method to twist the antipode of a bialgebroid using 2-cocycles, filling a gap in the understanding of antipode twisting beyond Hopf algebras.

Findings

A conjugation formula for the twisted antipode is valid under certain invertibility conditions.

The paper proves the existence of an invertible antipode for the twisted bialgebroid.

The approach generalizes antipode twisting from Hopf algebras to bialgebroids.

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.