Article

# Bosonization for dual quasi-bialgebras and preantipode

(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).

### Full-text preview

Available from: ArXiv
• Source
##### Article: Quantum Lines for Dual Quasi-Bialgebras
[Hide abstract]
ABSTRACT: In this paper, the theory to construct quantum lines for general dual quasi-bialgebras is developed followed by some specific examples where the dual quasi-bialgebras are pointed with cyclic group of points.
Algebras and Representation Theory 12/2013; 18(1). DOI:10.1007/s10468-014-9478-7 · 0.54 Impact Factor
• Source
##### Article: Quasi-Quantum Planes and Quasi-Quantum Groups of Dimension $p^3$ and $p^4$
[Hide abstract]
ABSTRACT: The aim of this paper is to contribute more examples and classification results of finite pointed quasi-quantum groups within the quiver framework initiated in \cite{qha1, qha2}. The focus is put on finite dimensional graded Majid algebras generated by group-like elements and two skew-primitive elements which are mutually skew-commutative. Such quasi-quantum groups are associated to quasi-quantum planes in the sense of nonassociative geomertry \cite{m1, m2}. As an application, we obtain an explicit classification of graded pointed Majid algebras with abelian coradical of dimension $p^3$ and $p^4$ for any prime number $p.$
Proceedings of the American Mathematical Society 01/2014; 143(10). DOI:10.1090/S0002-9939-2015-12602-0 · 0.68 Impact Factor
• Source
##### Article: On the Structure Theorem for Quasi-Hopf Bimodules
[Hide abstract]
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.
Applied Categorical Structures 01/2015; DOI:10.1007/s10485-015-9408-9 · 0.69 Impact Factor