Equivalent crossed products and cross product bialgebras

TL;DR

This paper characterizes when crossed product bialgebras are equivalent to certain tensor product structures, extending previous invariance results and providing a framework for understanding their equivalences.

Contribution

It proves a converse to a prior invariance result, offering a new characterization of equivalent crossed products and their relation to cross product bialgebras.

Findings

Characterization of equivalent crossed products.

Conditions under which cross product bialgebras are equivalent to tensor product structures.

Extension of invariance under twisting results.

Abstract

In a previous paper we proved a result of the type "invariance under twisting" for Brzezinski's crossed products. In this paper we prove a converse of this result, obtaining thus a characterization of what we call equivalent crossed products. As an application, we characterize cross product bialgebras (in the sense of Bespalov and Drabant) that are equivalent (in a certain sense) to a given cross product bialgebra in which one of the factors is a bialgebra and whose coalgebra structure is a tensor product coalgebra.