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Then in [2], [3], [4] and [5], the authors have discussed the structures of twisted Hopf algebras and bitwisted Hopf algebras and studied some properties of their antipodes, and so on. 在文献[2]、[3]、[4]和[5]中;作者分别研究了扭Hopf代数和双扭Hopf代数的结构和反极元的性质等.;本文主要研究双扭Hopf代数的对偶空间;以及两个双扭Hopf代数的对偶关系
bitwisted Hopf algebra 双扭Hopf代数
Last, the role of weak antipodes in the structures of modules/ comodules over a weak Hopf algebra is presented. 最后给出的一些结果体现了弱对极在弱 Hopf代数上的模 /余模结构中的作用 .
In addition, if B=H is a Hopf algebra with antipodes, then H0 is a Hopf algebra with antipodes S . 而且，如果B=H是一个带有对极S的Hopf代数，那么H~0是一个带有对极S~*的Hopf代数。
Some basic conclusions and their improper proofs in "Hopf Algebra"(E.Abe) are amended and improved in this paper. 本文对专著“Hopf Algebra”(E.;Abe)中若干基本结论或其证明的不妥或错误之处进行了修正和改进
In the second part of this thesis we prove that if the Hopf Galois extension is cocentral, and the invariants is a Hopf algebra then $A$ is itself a Hopf algebra. 设%24H%24是域%24k%24上的余半单Hopf代数，%24A%24为双代数且%24H%24在%24A%24上有余作用，在论文的第二部分我们证明了当%24A%24是其余不变量的Galois余中心扩张时，%24A%24上有Hopf代数结构当且仅当它的余不变量子双代数是 Hopf代数。
A well-known example is weak Hopf algebra, which is introduced in [L2] for studying the non-invertible solution of Yang-Baxter Equation based on this class of bialgebras (in [L2] and [L6]). 在文献[L2]中，为了研究Yang-Baxter方程的非平凡解，作者引入了著名的弱Hopf代数的概念使得基于这一类双代数，能够给出Yang-Baxter方程的一个非平凡解。
Proposition 2.8 Let H be a Hopf algebra and U a subHopfalgebra of H0 such that both H and U have bijective antipodes, and assume that U satisfies the RL-condition with respect to H. Let A be a U-comodule algebra, so that A is an H-module algebra as above. 命题2.;8 H是一个Hopf代数，U是H~0的一个子Hopf代数使得H和U有双射的对极，假定U关于H满足RL-条件。如上所述，A是一个U-余模代数，使得A是一个H-模代数。
In this paper,(A,SA) and(H,SH) both are weak Hopf algebras on a field k. A is also an H-module algebra. (A,SA)和(H,SH)都是数域k上的弱Hopf代数,并且A是右H-余模代数。
semiquasitriangular weak Hopf algebra 半拟三角弱Hopf代数
Let L and A be Hopf algebras on field k with antipodes SL and SA,and let C be a right A-module coalgebra. 设L是域k上的Hopf代数,其对极为SL;
Then, some regular semigroups were constructed by using of weak antipodes of some weak Hopf algebras. 然后用某些弱 Hopf代数的弱对极构造正则半群 .
Modern algebras, Homological algebras, Representation theory of algebras, Lie algebras, Hopf algebras. 教课领域：近世代数, 同调代数, 代数表示, 李代数，Hopf代数；
Let L and A be Hopf algebras on field k with antipodes s_L and s_A, B being a right A-comodule algebra, C a right A-module coalgebra. The defination of (A, B)-Hopf module and the fundamental structure theorem on (A, B)-Hopf module in L_LYD were given. 设L是域k上的Hopf代数;其对极为sL; A是-Hopf代数;其对极为sA;B是右A余模代数;C是右A模余代数;给出LLYD中(A;B)Hopf模的定义以及LLYD中(A;B)-Hopf模的基本结构定理;并讨论了其对偶情况.
NI Shen-bing.Weak Hopf algebras's twisted Smash product and Smach coproduct [J].J of Zhejiang University (Science Edition), 2002, 29(3): 268-273. [8]倪沈冰.;弱Hopf代数的扭曲的Smash积及Smash余积[J]
The Structure of Crossed Coproduct Bialgebra and Hopf Algebra 交叉余积双代数结构和Hopf代数
On the antipode and group-like elements of a braided Hopf algebra 辫子Hopf代数上的对极与类群元
The boy can't figure out the algebra problems. 这个男孩做不出这道代数题。
We've learned algebra in middle school. 我在中学就学过代数学。
She was studying to build up her algebra. 她在学习以提高自己的代数水平。