M < Sub > 3 < / Sub > (k) Of Ks < Sub > 3 < / Sub > - Die Algebraic Structure

The theory of group actions and Hopf algebra actions on algebras is an important research topic in algebra, and many mathematicians are working on the topic. In1976, Beattie introduced the concept of Hopf algebra actions on algebras in [1,2]. In1985, Blattner and Montgomery studied a duality theorem for Hopf module algebras in [3], which generalized the corresponding theorem of group action. Then many mathematicians were engaged in the theory of Hopf algebra action. For instance, Bergen and Cohen discussed the actions of commutative Hopf algebras in [4], and Cohen and Fishman studied general Hopf algebra actions in [5] etc. Chen and Zhang discribed the structure and classification of4-dimensional Yetter-Drinfeld module algebras over Sweedler4-dimensional Hopf algebra H4. In particular, they classified the H4-module algebra structures of the full matrix algebra M2(k). In1993, Montgomery summarized systematically the achievement on Hopf algebra action up to then in [18].Based on the above background, we discuss the isomorphism classification of kS3-module algebra structures of the full matrix algebra M3(k), where S3is the symmetric group on three elements. The thesis is organized as follows. In Section1, we recall some basic notions, such as module algebras over a Hopf algebra, group actions on algebras, module algebra isomorphism and so on. We also introduce the relations between these concepts, as well as some known conclusions for later use. Section2is the main part of the thesis, in which we first discuss the relation of weak similarity on pairs of matrices over Mn(k)×Mn(k), and describe the relation between the actions of kS3on the full matrix algebra Mn(k) and the weak similarity relationships on Mn(k)×Mn(k). It is shown that the classification of kS3-module algebras structure on the full matrix algebra Mn (k) is equal to the equivalent classification of the weak similarity on the set {(X1X2)|X1,X2∈GLn(k),X12∈kIn,X23∈kIn,{X1X2)2∈kln}. Then, based on reference [17], we give a set of representatives of equivalence classes with respect to the weak similarity relation for the sets of {(X1,X2)|X1,X2∈GL3(k),X12∈kI3,X23∈kI3,(X1X2)2∈kI}. Finally,we describe the kS3-module algebra structures of M3(k)and classify them by summarizing the discussion above.