Structure Of A Class Of 5-dimensional 3-Lie Algebras

An n-Lie algebra is a natural generalization of the concept of a Lie algebra to the case where the fundamental multiplication operation is n-ary n ≥ 2 (When n = 2, the definition agrees with the usual definition of a Lie algebra). Filippov[1] introduced the concept of an n-Lie algebra and investigated the structure of (n+l)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero and gave the multiplication tables. Kasymov[2J discussed the solvability and nilpotency of n-Lie algebras.In this paper, we investigate the structure of a class of 5-dimensional 3-Lie algebras Awhich contains a 4-dimensional subalgebra A₀ satisfying A¹ (?) A₀ over an algebraicallyclosed field of characteristic zero, and give the multiplication tables. Then we discuss the solvability of this class of 5-dimensional 3-Lie algebras. We also prove that the case ofdim A¹ = 5 does not exist.In section 1, we recall some definitions and notations of the theory of n-Lie algebras, such as the definitions of an n-Lie algebra, derived algebra of an n-Lie algebra, k-solvability and so on. In addition, we recall the classification theorem of (n+1)-dimensional n-Lie algebras over an algebraically closed field of characteristic zero.In section 2, we give the multiplication tables of a class of 5-dimensional 3-Lie algebrasA which contains a 4-dimensional subalgebra A₀ satisfying A¹(?)A₀, over an algebraicallyclosed field of characteristic zero. We also prove that the case of dim A¹= 5 does not exist.In section 3, from the multiplication tables of 5-dimensional 3-Lie algebras given in section 2, we discuss the solvability of this class of 5-dimensional 3-Lie algebras.