Cross Product Bialgebras - Part I

TL;DR

This paper develops a universal theory for cross product bialgebras without co-cycles, unifying known types and introducing new families within braided monoidal categories, with applications to Hopf bimodules.

Contribution

It provides a universal characterization of cross product bialgebras using projections and injections, and introduces Hopf data for a modular description, extending the theory to new bialgebra families.

Findings

Unified description of biproduct, double cross product, and bicross product bialgebras

Introduction of Hopf data for modular classification of cross product bialgebras

Recovery of known and new cross product bialgebras in braided categories

Abstract

The subject of this article are cross product bialgebras without co-cycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families of (co-cycle free) cross product bialgebras. Besides the universal characterization we find an equivalent (co-)modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majid's double biproduct is seen to be a twisting of a certain tensor product bialgebra in this category. This resembles the case of the Drinfel'd double which can be constructed as a twist of a specific cross product.