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数学进展 - 2018, Vol. 47(3): 401-412
研究论文
Majid double双积的一种广义型
A Generalization of Majid's Double Biproduct

马天水, 刘琳琳
MA Tianshui*, LIU Linlin

河南师范大学数学与信息科学学院, 新乡, 河南, 453007
School of Mathematics and Information Science, Henan Normal University, Xinxiang, Henan, 453007, P. R. China

收稿日期: 2016-10-05
出版日期: 2018-06-08
2018, Vol. 47(3): 401-412
DOI: 10.11845/sxjz.2016115b


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摘要 设$H$为双代数. $\sigma: H\otimes H\to A$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. $\tau: H\otimes H\to B$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$A\#_{\sigma} H_{\tau}\# B$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]).
关键词 交叉积Radford双积double双积    
Abstract:Let $H$ be a bialgebra. Let $\sigma: H\otimes H\to A$ be a linear map, where $A$ is a left $H$-comodule coalgebra, and an algebra with a left $H$-weak action. Let $\tau: H\otimes H\to B$ 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 $A\#_{\sigma} H_{\tau}\# B$ and the two-sided smash coproduct coalgebra $A\times H\times B$ to 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 Wang-Jiao-Zhao's crossed product in [Comm. Algebra, 1998, 26(4): 1293-1303] are all recovered from this.
Key wordscrossed product    Radford's biproduct    double biproduct
PACS:  O153.3  
通讯作者: E-mail: $*$ matianshui@yahoo.com   
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