Third Order The Full Matrix Algebra The S <sub> 3 </ Sub> - Graded

Matrix is an important mathematical concept, which also is an important tool in mathematical research. Matrix has a wide range of applications. For example, it is an attractive tool for computer scientists and cybernetics scientists. On the other hand, the group graded algebras, in particular, group graded algebra structures on a matrix algebra is one of important topics in algebra research. Many mathematicians are engaged in this research area. For example, Dascalescu, Ion, Nastasescu and Montes studied the good group-gradings on full matrix algebra in the literature [12]. Boboc and Dascalescu studied the gradings of matrix algebras by cyclic groups in [13]. Bahturin, Sehgal and Zaisev described the gradings of the matrix algebra over an algebraically closed field by abelian groups in [14]. For a given group grading on a full matrix algebra, Bahturin and Zaisev also decomposed such a graded algebra into a tensor product of graded subalgebras in [15]. Nastasescu and Oystaeyen summarized systematically the group gradings of matrix algebras in [2].In this master thesis, we mainly discuss the S 3-graded algebra structures of the full matrix algebra ( )M 3k , where S 3 is the symmetric group on three elements. The paper is organized as follows. In Section 1, we recall some notions such as module algebra, comodule algebra over a Hopf algebra, group actions and group grading of algebras. We also introduce the relation between these concepts, as well as some known conclusions. In Section 2, we first discuss the weak similarity relationships of 3×3-matrices over an arbitrary field, and give the representatives of equivalence classes with respect to the weak similarity relation for the sets of all 3×3invertible matrices, whose square or cubic are scalar matrices, respectively. Then we give all isomorphic classes of kC 2-module algebra structures and kC3 -module algebras structures of ( )M 3k . In Section 3, assuming that the characteristic of the field k is not equal to 2 and 3 and that k contains a primitive 3 th root of unity. Firstly, we describe the isomorphic classes of C 2-graded algebra structures and C3 -graded algebra structures of the full matrix algebra ( )M 3k by using the results given in Section 2. Then by refining the C 2-gradings and C3 -gradings of ( )M 3k , we get the isomorphic classes of all S 3-graded algebra structures of ( )M 3k .