ON CROSSED DOUBLE BIPRODUCT
ON CROSSED DOUBLE BIPRODUCT
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DOI：
10.1142/s0219498812502118
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发表时间：
2013-05
影响因子：
0.8
通讯作者：
Tianshui Ma;Zhengming Jiao;Yanan Song
中科院分区：
文献类型：
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作者：
Tianshui Ma;Zhengming Jiao;Yanan Song
设H是一个双代数。设σ：H <$H → A是一个线性映射，其中A是一个左H-余代数，且是一个具有左H-弱作用的代数.设B是右H-模代数，也是余模余代数.本文给出了单侧交叉积代数A#σ H # B与双侧smash余积代数A × H × B构成双代数的充要条件，称之为交叉双双积.马吉德的双重双积是从这个恢复。给出了具有Smash余积余代数结构的Brzezinski交叉积是双代数的充要条件。著名的拉德福的双积在[具有投影的霍普夫代数的结构，J.代数92（1985）322-347]中，统一的产品由Agore和Militaru在[扩展结构II：量子版本，J. Algebra 336（2011）321-341]和Wang-Jiao-Zhao的交叉积[Hopf algebra structures on crossed products Comm.Algebra26（1998）1293-#30300;都是特殊情况。
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 B be a right H-module algebra and also a comodule coalgebra. In this paper, we provide necessary and sufficient conditions for the one-sided crossed product algebra A#σ H # B and the two-sided smash coproduct coalgebra A × H × B to form a bialgebra, which we call the crossed double biproduct. Majid's double biproduct is recovered from this. Moreover, necessary and sufficient conditions are given for Brzezinski's crossed product equipped with the smash coproduct coalgebra structure to be a bialgebra. The celebrated Radford's biproduct in [The structure of Hopf algebra with a projection, J. Algebra92 (1985) 322–347], the unified product defined by Agore and Militaru in [Extending structures II: The quantum version, J. Algebra336 (2011) 321–341] and the Wang–Jiao–Zhao's crossed product in [Hopf algebra structures on crossed products Comm. Algebra26 (1998) 1293–1303] are all derived as special cases.