The Structure Theorem for quasi-Hopf bimodules: from quasi-antipodes to preantipodes

It is known that the Fundamental Theorem of Hopf modules can be used to characterize Hopf algebras: a bialgebra H over a field is a Hopf algebra (i.e. it is endowed with a so-called antipode) if and only if every Hopf module M over H can be decomposed in the form M coH ⊗ H , where M coH denotes the space of coinvariant elements in M . A partial extension of this equivalence to the case of quasi-bialgebras is due to Hausser and Nill: if a quasi-bialgebra admits a quasi-antipode, then it is possible to define a suitable space of coinvariants such that every quasi-Hopf bimodule could be decomposed in the same way. The main aim of this talk is to show how this characterization could be fully extended to the framework of quasi-bialgebras by introducing the notion of (the) preantipode for a quasi-bialgebra and by proving a Structure Theorem for quasi-Hopf bimodules. As a consequence some previous results, as the Fundamental Theorem of Hopf modules and the Hausser-Nill theorem for quasi-Hopf algebras, can be deduced from our Structure Theorem. This talk is based on the paper [P. Saracco, On the Structure Theorem for Quasi-Hopf Bimodules, preprint. (arXiv:1501.06061)].