POINTED HOPF ACTIONS ON FIELDS, I
POINTED HOPF ACTIONS ON FIELDS, I
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DOI：
10.1007/s00031-015-9328-7
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发表时间：
2014-03
影响因子：
0.7
通讯作者：
P. Etingof;Chelsea M. Walton
中科院分区：
文献类型：
--
作者：
P. Etingof;Chelsea M. Walton
半单Hopf代数在交换域上特征为零的代数闭域上的作用最近由作者在[18]中进行了分类。答案被证明是非常简单的--如果动作是内在忠实的，那么H一定是一个群代数。本文讨论的是非半简单情形，它要复杂得多。也就是说，我们研究了有限维(不一定是半单)Hopf代数在交换域上的作用，特别是当H是有限Cartan型点时，工作开始于H在域上的内部忠实作用的情况；这样的Hopf代数被称为Galois-理论。我们给出了这类Hopf代数的例子，其中包括Taft代数、Uq(2)以及其他小量子群的一些Drinfeld扭曲。我们还给出了许多非Galois理论的有限维Hopf代数的例子。有限维点Galois-理论Hopf代数的分类结果将在本研究的第二部分给出。
Actions of semisimple Hopf algebrasHover an algebraically closed field of characteristic zero on commutative domains were classified recently by the authors in [18]. The answer turns out to be very simple–if the action is inner faithful, then H has to be a group algebra. The present article contributes to the non-semisimple case, which is much more complicated. Namely, we study actions of finite dimensional (not necessarily semisimple) Hopf algebras on commutative domains, particularly when H is pointed of finite Cartan type.The work begins by reducing to the case where H acts inner faithfully on a field; such a Hopf algebra is referred to as Galois-theoretical. We present examples of such Hopf algebras, which include the Taft algebras, uq(2), and some Drinfeld twists of other small quantum groups. We also give many examples of finite dimensional Hopf algebras which are not Galois-theoretical. Classification results on finite dimensional pointed Galois-theoretical Hopf algebras of finite Cartan type will be provided in the sequel, Part II, of this study.