On 3-Lie algebras with abelian ideals and subalgebras

In this paper, we study the maximal dimension α(L) of abelian subalgebras and the maximal dimension β(L) of abelian ideals of m-dimensional 3-Lie algebras L over an algebraically closed field. We show that these dimensions do not coincide if the field is of characteristic zero, even for nilpotent 3-Lie algebras. We then prove that 3-Lie algebras with β(L)=m-2 are 2-step solvable (see definition in Section 2). Furthermore, we give a precise description of these 3-Lie algebras with one or two dimensional derived algebras. In addition, we provide a classification of 3-Lie algebras with α(L)=dimL-2. We also obtain the classification of 3-Lie algebras with α(L)=dimL-1 and with their derived algebras of one dimension.