Classification Of Representations Of 4-lie Algebras

The n-Lie algebra is a generaliztion of the Lie algebra, which is an algebraic system with an n-ary multilinear operation. N-Lie Algebra has its back ground in mechanics and manifolds. So it is necessary to study the structure and its rep-resentations. In this paper we studied the finite diemnsional representations of 5-dimensional 4-Lie algebras over an algebraically closed field of characteristic zero,and proved charaeteristic properties of irreducible representations of 5-dimensional 4-Lie algebras when the dimension of derived algebras are 3,4 and 5. And we gave classifications of irreducible representations ofκ-semisimple 5-dimensional 4-Lie al-gebras forκ≥2.The paper consists of four sections. The back ground and development of n-Lie algebras is introduced in the Section 1. In the Section 2 we give some definitions and results of n-Lie algebras which are used in the paper. In the Section 3 we give the classification theorem and certification.