A Class of Double Crossed Biproducts

• Published in: Ukrainian mathematical journal Vol. 70; no. 11; pp. 1767 - 1776
• Main Authors: Ma, T. S., Li, H. Y., Dong, L. H.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.04.2019; Springer; Springer Nature B.V
• Subjects: Algebra; Analysis; Applications of Mathematics; Geometry; Mathematics; Mathematics and Statistics; Statistics
• ISSN: 0041-5995; 1573-9376
• DOI: 10.1007/s11253-019-01621-y

Let H be a bialgebra, let A be an algebra and a left H-comodule coalgebra, and let B be an algebra and a right H -comodule coalgebra. Also let f : H ⨂ H → A ⨂ H , R : H ⨂ A → A ⨂ H , and T : B ⨂ H → H ⨂ B be linear maps. We present necessary and sufficient conditions for the onesided Brzeziński’s crossed product algebra A # R f H T # B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which generalizes the main results from [“On Ranford biproduct,” Comm. Algebra, 43 , No. 9, 3946–3966 (2015)]. It is clear that both the Majid double biproduct [“Doublebosonization of braided groups and the construction of U q ( g ), ” Math. Proc. Cambr. Philos. Soc., 125 , No. 1, 151–192 (1999)] and the Wang–Jiao–Zhao crossed product [“Hopf algebra structures on crossed products,” Comm. Algebra, 26 , 1293–1303 (1998)] are obtained as special cases.