The Classification Of N-lie Algebras With Special ?(L)

In the study of algebraic structure,classification is one of the most important,and most difficult problems.In this paper,the structure of non-commutative n-Lie algebras over a field F of characteristic zero,which satisfies ?(L)=m-n and ?(L)=m-n+1 is classified,where ?(L)is the largest dimension of abelian ideals of a non-commutative n-Lie algebra L.First,the structure of n-Lie algebras with ?(L)=m-n+1 is studied,and n-Lie algebras satisfying dim L1<4 are classified.It is proved that if L satisfies Z(L)(?)L1,then m-n+1/2?dim L1 ? m-n+1,and in the cases dim L1=1,2 and 3,there are 2,6 and 11 classes n-Lie algebras,respectively;furthermore,if Z(L)(?)L1,then the dimension of L satisfy n?dim L ? n+1,n+1 ? dim L ? n+3,and n+2 ? dim L ? n+5,respectively.Finally,the structure of n-Lie algebras with ?(L)=m-n is studied,and 3-Lie algebras that satisfy dim L1<3 are classified.It is proved that L is nilpotent if and only if there is a nilpotent subalgebra A satisfying dim A1=dim L1,and in the cases dim L1=1 and 2,there are 2 and 24 classes 3-Lie algebras respectively;in addition,if Z(L)(?)L1,then the dimension of L satisfies 6 ? dim L ? 7,and 5 ? dim L ? 11,respectively.