Algebraic Structures Of3-Indices Matrix Space

The paper mainly concerns multiplicative structures of3-indices matrix (cubic ma-trix) and algebraic structures of3-indices matrix space. Fifteen kinds of multiplicationsT M(r.s) for arbitrary two cubic matrices which satisfying the associative law are pro-vided, and the relationship between diferent multiplications discussed. For an N-ordercubic matrix A, three kinds of determinant|A|s, s=1,2,3and four kinds of”trace”At, t=1,2,3,4are definied. According to these multiplications T M(r.s), the Liealgebra Lrsand3-Lie algebra Jrsare constructed, respectively, and the structures arestudied.Throughout the paper, the characteristic of a field F is assumed to be zero. Theorganization of the paper is as follows:Section1introduces the background and the development of matrix.Section2defines15kinds of multiplications which satisfy the associative law.Section3defines the determinants and traces of the cubic matrix.Section4constructs of Lie algebras by cubic matrix.Section5constructs of3-Lie algebras by cubic matrix.