Smash H_n(1,q)-extended Algebra

The extension of algebras is one of the major skills for us to study algebras,which is usually used to research the structures and classifications of algebras.And the important contents in algebra include the theory of group action and Hopf algebra action.Numbers of scholars have been studying the theory,and using the theory of Hopf algebra action to the studying of algebras,extensions and structures.Cleft extension for a Hopf algebra amount to the crossed product of the Hopf algebra.The concept extends the Galois extension of a group with a normal basis,and equivalently extends the concept of crossed product of group actions.The ideal of cleft extension and crossed product unify and extend those of smash extension and smash product,and twisted extension and t.wisted product.There have been abundant relevant research.Taft algebra is a kind of pointed Hopf algebra,which plays an important role in the research of the structures and classifications of Hopf algebra.And the Hopf algebra Hn(1,q)is exactly isomorphic to the Drinfeld double of the Taft algebra.The representation category of Hn(1,g)has braided structure,and it can prove the solution of the quantum Yang-Baxter equation.There has been many interesting research result revolving around the representation theory and the tensor product structure of the representation category with finite dimension of Hn(1,q).However,few achievements have been made on the subjects about algebraic Hopf Galois extension,cleft extension,smash extension and so on,which are depended on Hn(1,q).In this paper,we research the smash Hn(1,q)-extension of algebras based on the per-existing work.The thesis is divided into two parts.In the first part of the article,we introduce the fundamental definitions of the Hopf algebra,comodule algebra,H-extension of algebras depended on the Hopf algebra H,cleft H-extension,smash H-extension and so on,and recommend some essential properties and relevant conclusions about cleft H-extension,smash H-extension.Meanwhile we introduce the structures and elementary properties of the skew group algebra and algebraic Ore extension.The second section is the principal part of the paper Firstly,it presents the structures of the Hopf algebra Hn(1,q),then we make use of R-point of Hn(1,q)to give a equivalent characterization about a Hn(1,q)-extension R0(?)R to become a smash extension.So,we can get a Hn(l,q)-smash system(R,?)over R0.And then,this smash system can derive a group of linear transformations of R0.We describe the properties of this group of linear transformations.The group of linear transformations over It like this is called a Hn(1,q)-smash datum,then we prove that this datum can make R0 to become a left Hn(l,q)-module algebra.Secondly,for any given Hn(1,q)-smash datum d over R0,we make use of the definition of the skew group algebra and Ore extension to give a general method to eonstruct an extension Rd over Ro.We prove that Rd is a Hn(1,q)-smash extension of R0,and then give some basic properties of Rd.From this,we prove that any Hn(1,q)-smash extension of R0 is isomorphic to some Rd.Finally,for any two Hn(1,q)-smash datum d,d'over R0,we give the equivalent characterization when Rd,is isomorphic to Rd,which are both Hn(1,q)-extension of R0.