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On the Structure Theorem for quasi-Hopf bimodules

Articolo
Data di Pubblicazione:
2017
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 $M^mathrmcoHotimes H$, where $M^mathrmcoH$ 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.
Tipologia CRIS:
03A-Articolo su Rivista
Keywords:
Quasi-bialgebras, structure theorem, preantipode, right quasi-Hopf bimodules, quasi-Hopf algebras, gauge
Elenco autori:
Saracco, P.
Link alla scheda completa:
https://iris.unito.it/handle/2318/1522188
Link al Full Text:
https://iris.unito.it/retrieve/handle/2318/1522188/41098/Saracco%20-%20On%20the%20structure%20theorem%20for%20quasi-Hopf%20bimodules.pdf
Pubblicato in:
APPLIED CATEGORICAL STRUCTURES
Journal
  • Dati Generali

Dati Generali

URL

http://arxiv.org/abs/1501.06061v1
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