A class of double crossed biproducts
AbstractLet $H$ be a bialgebra, let $A$ be an algebra and a left $H$-comodule coalgebra, let $B$ be an algebra and a right $H$-comodule coalgebra. Also let $f : H \otimes H \rightarrow A \otimes H, R : H \otimes A \rightarrow A \otimes H$, and $T : B \otimes H \rightarrow H \otimes B$ be linear maps. We present necessary and sufficient conditions for the one-sided Brzezi´nski’s crossed product algebra $A\#^f_RH_T\#B$ and the two-sided smash coproduct coalgebra $A \times H \times B$ to form a bialgebra, which generalizes the main results from [On Ranford biproduct // Communs Algebra. – 2015. – 43, № 9. – P. 3946 – 3966]. It is clear that both Majid’s double biproduct [Double-bosonization of braided groups and the construction of $U_q(g)$ // Math. Proc. Cambridge Phil. Soc. – 1999. – 125, № 1. – P. 151 – 192] and the Wang – Jiao – Zhao’s crossed product [Hopf algebra structures on crossed products // Communs Algebra. – 1998. – 26. – P. 1293 – 1303] are obtained as special cases.
How to Cite
Dong, L. H., H. Y. Li, and T. S. Ma. “A Class of Double Crossed Biproducts”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 70, no. 11, Nov. 2018, pp. 1533-40, http://umj.imath.kiev.ua/index.php/umj/article/view/1657.