设H是一个双代数,B是带有H弱作用的代数,σ:H(?)H→B和τ:H(?)H→B都是k-双线性映射.首先我们给出了B_χ~(#_σ~τ)H成为双代数的充分必要条件,此双代数带有扭曲交叉积B#_σ~τH和冲余积B×H,其中B是H上的余模余代数.此双代数是由Radford首次在文献[8]中提出,后来Doi and Takeuchi又在文献[4]和[9]中进一步推广而得到的.然后我们对此双代数进行刻画并研究其基本性质.最后我们给出了此双代数成为Hopf代数的充分条件.
Let H be a bialgebra.Let H act weakly on an algebra B,σ:H(?)H→B be a k-bilinear map and letτ:H(?)H→B be a k-bilinear map.Then we find a sufficient and necessary condition for B_×~(#_σ~τ)H,with the twisted crossed product B#_σ~τH and smash coproduct B×H for a comodule coalgebra B over H,to form a bialgebra, which generalizes the ones introduced by Radford,Doi and Takeuchi in[4]and[9]. Next we obtain a characterization for this new bialgebra and study its basic properties. In addition,we derive a sufficient condition for this new bialgebra to be a Hopf algebra.