Hopf algebra smash product and Quantum Weyl algebra

1. P-伽罗瓦扩张的概念是由K.岸本这个概念包含了通常的伽罗瓦扩张，纯不可分扩张和Hopf伽罗瓦扩张。我们确定了域上除了P是循环群之外的所有三次P-Galois扩张。P-Galois扩张的同构类是否具有群结构还不清楚。在我们的例子中，同构类不能很好地工作。2.设A是环和M-双模。设f是从A到M的可加映射，ω是M中的元素.（f，ω）称为= f（x）y+xf（y）+xωy（x，y ∈ A）的广义导子.（f，ω）是Bresar广义导子，若A有单位元，则它们相等.给出了导子、广义导子和Bresar导子的初等关系，并确定了广义导子集gDer（A，M）与从A到M的导子集Der（A，M）之间的函数关系.利用这个结果，我们给出了关于M，gDer（A，M）和Der（A，M）的可裂短正合列.此外，给出了广义导子的泛映射性质。将广义导子的概念推广到Jordan导子和Lie导子，得到了与广义Jordan导子类似的结果。本文还讨论了广义高阶导子，从Hopf代数的观点出发，广义高阶导子的作用是余代数作用。因此，我们有一个应用程序，这些地图的霍普夫伽罗瓦理论。

1. The notion of P-Galois extensions was introduced by K. Kishimoto. This notion contains that the usual Galois extensions, purely inseparable extensions and Hopf Galois extensions. We determine all cubic P-Galois extensions over a field except that P is a cyclic group. It is not known that the isomorphism classes of P-Galois extensions has a group structure or not. In our case, the isomorphism classes does not work well.2. Let a be a ring and an M-bimodule. Let f be an additive map from A to M and ω an element in M. (f, ω) is called a generalized derivation of = f(x)y+xf(y)+xωy (x, y ∈ A). (f, ω) is a Bresar's generalized derivation and if A has an identity, they are equal. We give elementary relations of derivations, generalized derivations and Bresar's one and determine a functional relation between gDer(A, M), the set of all generalized derivations and Der(A, M), the set of all derivations from A to M. Using this result, we give a split short exact sequence with respect to M, gDer(A, M) and Der(A, M). Moreover, the universal mapping property for generalized derivations is given. The notion of generalized derivation is extended to Jordan and Lie derivations and we can get similar results to generalized Jordan derivations. Generalized higher derivations are also discussed.In the stand point of Hopf algebras, the action of generalized derivation is a comodule algebra action. Therefore we have an application of these maps to Hopf Galois theory.