Structures Of 3-Lie Algebras Constructed By 2-Cubic Matrix

The concept of n-Lie algebra was introduced by V. T. Filippov in 1985, the multi-plication was generalized from Lie product to the n-ary multiplication. It looks like that the n-Lie algebra is the usuall generalization of Lie algebra, but n-Lie algebra, especially, 3-Lie algebra, has wide applications in mathematics, mathematical physics and string theory.We know that the realization of 3-Lie algebras is very important in the structureal theory of 3-Lie algebras all the time. The 3-Lie algebras can be realized by Lie algebras and their functions, also by commutative associative algebras, Pre-Lie algebras and theire derivations. In this paper, the 3-Lie algebras are realized by 2-cubic matrix over the prime field Z2 of characteristic 2. First the multiplications*11 and*21 of 2-cubic matrix are defined, and by the trace of cubic matrix, two 8-dimensional 3-Lie algebras Γ11 and F21 are constructed. The structure of Γ11 and Γ21 and inner derivation algebras ad(Γ11) and ad(Γ21) are studied, and the concrete expression of all the inner derivations are provided.The paper consists of five sections. Section 1 introduces the back ground and de-velopment of 3-Lie algebras. Section 2 gives some definitions and some results which are used in the paper. Section 3 constructs 3-Lie algebra Γ11 and studies its strcutures. Section 4 constructs 3-Lie algebra Γ21 and studies its strcutures. Section 5 discusses the structures of inner derivation algebras ad(Γ11) and ad(Γ11), respectively.