Abstract
For a quasi-bialgebra H, we show that the category C of H-bimodules is duoidal and that the so called u-coaugmented bimonoids in C are exactly the quasi-bialgebras with a coalgebra projection. When H is a quasi-Hopf algebra with bijective antipode, we prove that the u-coaugmented bimonoids in C can also be described as what we will call double wreath quasi-Hopf algebras, objects determined by H and pre-bialgebras R within the category of Yetter-Drinfeld modules over H. A particular class of double wreath quasi-Hopf algebras is obtained by deforming with a 2-cocycle the multiplication of a Radford biproduct quasi-Hopf algebra. Other classes of this type are given by the symplectic fermion quasi-Hopf algebras.
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