中文摘要：

本文研究了一个双扭Hopf代数的分次对偶空间以及两个双扭Hopf代数的分次对偶关系．利用代数和余代数分次对偶空间的性质，得出一个局部有限的双扭（Х^1,Х2）-Hoof代数的分次对偶空间是一个双扭（Х1^T。Х2）-Hopf代数。并判定两个双扭Hopf代数的分次对偶可以简化为判定它们作为双扭双代数是分次对偶的．

英文摘要：

The aim of this paper is to discuss the gradedly dual space of a bitwisted Hopf algebra, and the gradedly duality relationship between two bitwisted Hopf algebras. By using the properties of the dual space of a graded algebra and coalgebra, we prove that the gradedly dual space of a local finite （Х1 , Х2 ）- bitwisted Hopf algebra is a （X1^T ,Х2 ）-bitwisted Hopf algebra, and then we point out that when we want to prove two bitwisted Hopf algebras to be graded dual as Hopf algebras, we only have to prove that they are graded dual as bitwisted bialgebras.