The 2-cocycle On The Nine-dimensional Taft Algebra

Hopf algebra is an important research field in algebra,which originated in 1940s.Hopf algebra is an algebra system possessing both an associative algebra structure and a coassociative coalgebra structure which was found by Heinz Hopf when he studied the topological properties of Lie groups.During the past more than 30 years,with the in-depth study of quantum groups,it was found that the quantum groups are some special Hopf algebra.The research of quantum groups and Hopf algebras crossed fusion and developed rapidly.Many research results on the structures and classification of Hopf algebra have been achieved.Many research methods appeared,for instance,2-cocycle deformation is an important technique.It is found that the Drinfeld double of finite dimensional Hopf algebra is exactly a deformation of a tensor product Hopf algebra by a 2-cocycles.In recent years,many researchers use 2-cocycles deformation to study the properties and the classification of Hopf algebra.Taft algebras are one class of important noncommutative and noncocommutative Hopf algebras,which play an important and ignored role in the study of the quantum groups and Hopf algebras.The generating relations of generator of Taft algebras are extensively drawn and used when some Hopf algebras and noncommutative algebras are constructed.During past several years,the structures,properties,representations and Green rings of Taft algebras and their Drinfeld doubles have been extensively studied,many interesting results have been achieved.Therefore,it is meaningful to do further study on the 2-cocycles over the Taft algebras.In this thesis,we will investigate the structures and classification of 2-cocycles on 9-dimensional Taft algebra H = H3(q)based on the predecessors' research about Taft algebra,here the involved 2-cocycles are always convolution-invertible.The thesis is organized as follows.In Section 1,we will make preliminaries,and some basic nations about Taft algebras,2-cocycles,Lazy 2-cocycles and related basic concepts.In Section 2,we will investigate the Lazy 2-cocycles on H.It is shown that a normal Lazy 2-cocycles on H is exactly determined by one parameter.Furthermore,we will prove that the multiplicative group Nm(H)of all normal Lazy 2-cocycles on H is isomorphic to the addition group of the ground field.Thus,the Lazy 2-cocycles on H are classified.In Section 3,we will investigate the 2-cocycles on H generally.In order to do so,we only need to determine all normal 2-cocycles.Since the right multiplication of the group Nm(H)on the set of all 2-cocycles on is a group action,the all 2-cocycles on H are divided into the intersected orbits.Hence with the help of the result obtained in Section 2,we only give a representative for each orbit.At first,we show that the representative of each orbit are the 2-cocycles with a special property.Then we show that by 2 nonzero parameters and 6 general parameters,one can construct such a representative element,i.e.,a 2-cocycle with given special property.Finally,we show that a representative element of an orbit can be always constructed by 8 parameters as before.Thus,we obtain the structures of all 2-cocycles on the 9 dimensional Taft algebra,and their classification.