OPERATORS ON HOPF ALGEBRAS.
OPERATORS ON HOPF ALGEBRAS.
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DOI：
10.2307/2374012
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发表时间：
1977-02
影响因子：
1.7
通讯作者：
D. Radford
中科院分区：
文献类型：
--
作者：
D. Radford
设A是域k上具有对极S的有限维Hopf代数。在(6)中，S和S2的结构被认为与A和A*的重要代数特征密切相关。本文的研究进一步深化了本文的研究。更一般地，考虑任意Hopf代数A的“局部有限”双代数自同态t：A->；A(这意味着A是有限维t-不变子空间的和)。给出了有限维Hopf代数的应用。我们证明了域上任意Hopf代数的对极S是内射(双射)的，如果它仅限于冠状代数时也是内射的。我们证明了，如果S是局部有限算子，或者如果它是有限维的，则它是双射的。分别用6阶和8阶对极构造了环Q(可定义在Z上)上的有限维Hopf代数。这些例子表明，对极的顺序和基场中的单位原根之间的关系并不像(6)中的一些结果那样错综复杂。
Let A be a finite dimensional Hopf algebra with antipode s over a field k. In (6) the structure of s and s2 is seen to be closely related to the important algebraic features of A and A*. This paper furthers the investigation. More generally "locally finite" bialgebra endomorphisms t:A ->A of an arbitrary Hopf algebra A are considered (this means A is the sum of the finite dimensional t-invariant subspaces). Applications are given to finite dimensional Hopf algebras. We show that the antipode s of an arbitrary Hopf algebra over a field is injective (bijective) if the same is true when it is restricted to the coradical. We prove that s is bijective if it is a locally finite operator, or if the coradical is finite dimensional. Examples of finite dimensional Hopf algebras over Q (which can be defined over Z) are constructed with antipode of order 6 and 8, respectively. These examples show that the relationship between the order of the antipode and the primitive roots of unity in the ground field is not as intricate as indicated by some of the results of (6).