Cleft E (2) - Expansion Of Algebra

The extension of algebras is an important method for studying algebras, which has been widely used to study the structures and classification of algebras. On the other hand, The theorey of group action and Hopf algebra action is one of the important topics in algebra. Many mathematicians have been studying the theorey, and applying it to the studying of algebra extensions and algebra structures. Cleft extension for a Hopf algebra is equivalent to the crossed product of the Hopf algebra. The concept extends the Galois extension of a group with a normal basis, or equivalently extends the concept of crossed product of group actions. The concepts of cleft extension and crossed product unify and extend those of smashed extension and smash product, and twisted extension and twist product. This is the expansion and in-depth study of the Hopf algebra actions, coactions and algebra extension theory.In this paper, we investigate the cleft extension of a given algebra by the Hopf algebra E(2), or equivalently, the crossed product of a given algebra with the Hopf algebra E(2). At first, we introduce the structure and related properties of the Hopf algebra E(2). Then for a given algebra C, we give a necessary and sufficient condition for an E(2)-extension of C to become a cleft extension, and give some important properties of a cleft E(2)-extension of C. Meanwhile, form the corresponding cleft system, we derive a datum and give the properties satisfied by the datum. Furthermore, we introduce the concept of an E(2)-cleft datum over an algebra C, and denote by D(E(2),C) the set of all E(2)-cleft data over C. Next, for any given E(2)-cleft datum d∈<D(E(2),C) over C, we construct an extension Ad of C. It is shown that Ad admits a universal property, and that Ad is a cleft E(2)-extension of C. We also give a necessary and sufficient condition for such two E(2)-extensions of C to be isomorphic. Meanwhile, it is shown that any cleft E(2)-extensions of C is isomorphic to some Ad, where d∈D(E(2),C). Using the set of orbits of a semidirect product group acting on D(E(2),C), we classify all cleft E(2)-extensions of C up to isomorphism. Finally, we discuss the necessary and sufficient conditions for Ad to become a twisted E(2)-extensions of C and a smashed E(2)-extensions of C, respectively, where d∈<D(E(2), C).