On Radford Biproduct

Abstract

Let H be a bialgebra, A an algebra, and a left H-comodule coalgebra. Let f: H ⊗ H → A ⊗ H and R: H ⊗ A → A ⊗ H be two linear maps. Let B be a right H-module algebra and also a right H-comodule coalgebra. In this paper, necessary and sufficient conditions are given for the one-sided Brzeziński's crossed product algebra and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the Brzeziński's double biproduct. The celebrated Radford biproduct [18], Majid's double biproduct [13], Agore and Militaru's unified product [2], and the Wang–Jiao–Zhao's crossed product [21] are all derived as special cases. On the other hand, we construct a class of Radford biproducts by a twisting method.

Key Words:

• Double biproduct
• Radford biproduct

2010 Mathematics Subject Classification:

• 16T05

ACKNOWLEDGMENTS

The authors are deeply indebted to the referee for his/her very useful suggestions and some improvements to the original manuscript.

Notes

Communicated by L. Ein.