E (2) On The BiGalois Object

In algebra, the extension of algebras is an important research method, which is used widely to study the structures and classification of algebras. On the other hand, the action theory of group and Hopf algebra is an important topic, which is used by many mathematicians to study the structures and extension of algebras. From the reference [4], one can see that any right Galois extension of k gives rise to a unique Hopf algebra over which A is a left Galois extension of k such that A becomes a bicomodule (called an biGalois object).In this paper, we mainly study the BiGalois objects over the Hopf algebra E(2). First, we introduce the structure and related properties of the Hopf algebra E(2), and then give an equivalent condition for a right E(2)-comodule algebra A to become a right E(2)-Galois object. Furthermore, a new description of a right E(2)-Galois object is given, and a series of important properties of the right E(2)-Galois objects are gotten. Meanwhile, by the corresponding cleft system, a set of parameters is derived, and the conditions satisfied by the parameters are given. It is shown that the set of all these parameters is exactly k x k x k x k, where k is the base filed, k* denote the multiplicative group of the nonzero element in k. Secondly, for any given parameter set d∈k*×k×k×k, we construct a right E(2)-Galois object Ad, and define a left E(2)-comodule structure λd. It is shown that Ad is also a left Galois object and is a bicomodule. Thus, Ad is a BiGalois object over E(2). Meanwhile, for two given parameter sets d,d’∈k*×k×k×k, the equivalent condition for Ad= Ad, is given. We also use the set of orbits of a semidirect product group acting on k* ×k×k×k to classify all E(2)-BiGalois objects. Finally, using the classification of E(2)-BiGalois objects, we describe the group BiGal(E(2)) of E(2)-BiGalois objects.