Two results on double crossed biproducts

Abstract

Let H be an algebra with a distinguished element ε H ∈ H * and C, D two coalgebras. Based on the construction of Brzeziński's crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra C × G T H β R ×D which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for C × G T H β R × D equipped with two-sided tensor product algebra C ⊗ H ⊗ D to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct C ⋆ α H β ⋆ D in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of C ⋆ α H β ⋆ D similar to Majid double biproduct C ⋆ H ⋆ D and also present some related structures.