Characteristics Of Algebraically Closed Field Of 2 4 D H < Sub > 4 < / Sub > - Die Algebraic Classification

During past decades, the theory of Hopf algebras and quantum groups has been a hot topic in algebra, which has a deap relationship with mathematical physics. H4-module algebras play an important role in the study of Hopf algebras and quantum groups, which has been studied tensively. On the other hand, the structures and classifications of various algebra systems are important research topics. In1975, Gabrel introduced the concept of geometric classification of algebras, and described the algebraic classification and geometric classification of4-dimensional algebras in [1]. From then on, the algebraic classification and geometric classification of some algebra structures, such as5-dimensional algebras and4-dimensional superalgebras, have been obtained. Chen and Zhang [6] gave the algebraic classification of4-dimensional Yetter-Drinfeld H4-module algebras over a field k with char(k)≠2in2006. In2009, Armour, Chen and Zhang described the algebraic classification of4-dimensional superalgebras in [7]. Based on these results, we investigate the algebraic classification of4-dimensional differential superalgebras, i.e.,4-dimensional H4-module algebras, over a field k with char(k)≠2.The thesis is organized as follows. In Section1, we recall some basic concepts which are used in this thesis, such as Hopf algebra, differential superalgebra, H4-module algebra and Yetter-Drinfeld H4-module algebra. We also simplify the classification of the4-dimensional superalgebras given in [7], and display a new description of these algebra structures. In Section2, based on the description of4-dimensional superalgebras given in Section1, we first consider the possible differential structures on these superalgebras such that they become differential superalgebras, i.e.,H4-module algebras. Thus we obtain some H4-module algebras. Then we study the isomorphism relationships among these H4-module algebras, and chose a representative element from each isomorphism class. Thus, a set of non-isomorphic H4-module algebras is given. In Section3, using the general representation theory of triangular Hopf algebras, we prove that any4-dimensional H4-module algebra must be isomorphic to some representative element of H4-module algebras given in Section2, by examining the 4-dimensional Yetter-Drinfeld H4-module algebras given in [7]. Thus, we complete the algebraic classification of4-dimensional differential superalgebras, i.e.,4-dimensional H4-module algebras.