Bosonization for dual quasi-bialgebras and preantipode

TL;DR

This paper develops a bosonization construction for dual quasi-bialgebras, characterizes those with a projection onto a dual quasi-bialgebra with a preantipode, and explores their graded coalgebra structures.

Contribution

It introduces a bosonization method for dual quasi-bialgebras and characterizes those with a projection onto a dual quasi-bialgebra with a preantipode.

Findings

Bosonization associates a dual quasi-bialgebra to each dual quasi-bialgebra and bialgebra in Yetter-Drinfeld modules.

Characterization of dual quasi-bialgebras with a projection onto a dual quasi-bialgebra with a preantipode.

Analysis of the graded coalgebra structure for dual quasi-bialgebras with the dual Chevalley property.

Abstract

In this paper, we associate a dual quasi-bialgebra, called bosonization, to every dual quasi-bialgebra $H$ and every bialgebra $R$ in the category of Yetter-Drinfeld modules over $H$. Then, using the fundamental theorem, we characterize as bosonizations the dual quasi-bialgebras with a projection onto a dual quasi-bialgebra with a preantipode. As an application we investigate the structure of the graded coalgebra $grA$ associated to a dual quasi-bialgebra $A$ with the dual Chevalley property (e.g. $A$ is pointed).