Majid double双积的一种广义型（英文）

A Generalization of Majid's Double Biproduct

设H为双代数.σ:HH→A为线性映射,其中A为左H余模余代数,且是带有左H-弱作用的代数.τ:HB→B为线性映射,其中B为右H余模余代数,且是带有右H-弱作用的代数.本文给出双边交叉积A#_σH_τ#B和双边smash余积构成双代数的充要条件.这一结构包括了著名的Radford双积(见[J.Algebra,1985,92(2):322-347]),Majid double双积(见[Math.Proc.Cambridge Philos.Soc.,1999,125(1):151-192]),以及王栓宏、焦争鸣和赵文正定义的交叉积(见[Comm.Algebra,1998,26(4):1293-1303]).

Let H be a bialgebra. Let σ ： HH→A be a linear map, where A is a left H-comodule coalgebra, and an algebra with a left H-weak action. Let τ： HH→B be a linear map, where B is a right H-comodule coalgebra, and an algebra with a right H-weak action. In this paper, we provide necessary and sufficient conditions for the two-sided crossed product algebra A#σHτ#B and the two-sided smash coproduct coalgebra A×H×Bto form a bialgebra. The celebrated Radford＇s biproduct in [J. Algebra, 1985, 92（2）： 322-347], Majid＇s double biproduct in [Math. Proc. Cambridge Philos. Soc., 1999, 125（1）： 151-192] and the WangJiao-Zhao＇s crossed product in [Comm. Algebra, 1998, 26（4）： 1293-1303] are all recovered from this.