Globalization for geometric partial comodules

Comment on ‘Twisted bialgebroids versus bialgebroids from a Drinfeld twist’

“…The component σ AA : [3] quasitriangularity is required in the statements, their proofs additionally use formula (4) for the H-coaction of any braided commutative YD H-module algebra. This is unsatisfactory and misleading as implied by the following simple counterexample provided to us by Saracco and Vercruysse [16].…”

“…However, while Z Š did suggest to avoid assuming quasitriangularity and using in deriving the main result the twist of the entire category of YD modules similarly to the logic shown above, Z Š failed to realize that the actual proof in [3] does not cover general scalar extension bialgebroids even for quasitriangular bialgebras and noticed this only as late as early 2022. We thank P Saracco and J Vercruysse for kind communication of counterexamples [16,21] and T Brzeziński for initiating that communication.…”

Geometric partial comodules over flat coalgebras in Abelian categories are globalizable

“…It can be shown (see [38,Lemma 3.2]) that if (Y, p) is a globalization of a partial comodule X, then p : Y → X is an epimorphism. Moreover, it follows from axiom (GL2) that a globalization of a partial comodule X is unique, whenever it exists.…”

“…Lead by the need of a unified approach to the vast panorama of the theory of partial actions and in order to tackle the globalization problem, in [38] we discussed the question from a category-theoretical perspective by taking advantage of the notion of geometric partial comodule introduced in [29]. We also provided a genuine procedure to construct globalizations (whenever they exist) that can be applied to many concrete cases of interest.…”

Partial and global representations of finite groups

“…By Remark 1.15, this result extends the ones in [ABV19] and [Aba18] (cf. also [SV20]), giving a construction that is closer to the one for partial actions in [Aba03].…”