Article

# Bosonization for dual quasi-bialgebras and preantipode

Journal of Algebra (Impact Factor: 0.6). 11/2011; 390. DOI: 10.1016/j.jalgebra.2013.05.014

Source: arXiv

**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).

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).

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**ABSTRACT:**The Structure Theorem for Hopf modules states that if a bialgebra $H$ is a Hopf algebra (i.e. it is endowed with a so-called antipode) then every Hopf module $M$ is of the form ${M}^{\mathrm{co}{H}}\otimes H$, where ${M}^{\mathrm{co}{H}}$ denotes the space of coinvariant elements in $M$. Actually, it has been shown that this result characterizes Hopf algebras: $H$ is a Hopf algebra if and only if every Hopf module $M$ can be decomposed in such a way. The main aim of this paper is to extend this characterization to the framework of quasi-bialgebras by introducing the notion of preantipode and by proving a Structure Theorem for quasi-Hopf bimodules. We will also establish the uniqueness of the preantipode and the closure of the family of quasi-bialgebras with preantipode under gauge transformation. Then, we will prove that every Hopf and quasi-Hopf algebra (i.e. a quasi-bialgebra with quasi-antipode) admits a preantipode and we will show how some previous results, as the Structure Theorem for Hopf modules, the Hausser-Nill theorem and the Bulacu-Caenepeel theorem for quasi-Hopf algebras, can be deduced from our Structure Theorem. Furthermore, we will investigate the relationship between the preantipode and the quasi-antipode and we will study a number of cases in which the two notions are equivalent: ordinary bialgebras endowed with trivial reassociator, commutative quasi-bialgebras, finite-dimensional quasi-bialgebras.