Two results on double crossed biproducts

Abstract

Let $H$ be an algebra with a distinguished element ${\epsilon }_H\in H^{\ast }$ and $C,D$ two coalgebras. Based on the construction of Brzeziński’s crossed coproduct, under some suitable conditions, we introduce a coassociative coalgebra $C\times {}_T^G{H}_R^{\beta }\times D$ which is a more general two-sided coproduct structure including two-sided smash coproduct. Necessary and sufficient conditions for $C\times {}_T^G{H}_R^{\beta }\times D$ equipped with two-sided tensor product algebra $C\otimes H\otimes D$ to be a bialgebra (Hopf algebra) are provided. On the other hand, we obtain an improved version of the double crossed biproduct $C\star {}^{\alpha }{H}^{\beta }\star D$ in [An extended form of Majid's double biproduct, J. Algebra Appl. 16 (4), 1760061, 2017] which induces a description of $C\star {}^{\alpha }{H}^{\beta }\star D$ similar to Majid double biproduct $C\star H\star D$ and also present some related structures.

Keywords

• crossed coproduct
• Radford biproduct
• double biproduct

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