In this paper we study the projections of weak braided Hopf algebras using the notion of YetterDrinfeld module associated with a weak braided Hopf algebra. As a consequence, we complete the study of the structure of weak Hopf algebras with a projection in a braiding setting obtaining a categorical equivalence between the category of weak Hopf algebra projections associated with a weak Hopf algebra H living in a braided monoidal category and the category of Hopf algebras in the non-strict braided monoidal category of left-left Yetter-Drinfeld modules over H.
In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, ϱA) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product AH♯σAHH, where AH is the coinvariants subalgebra of A and σAH is a morphism between H ⊗ H and AH. As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.
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