Two-step Nilpotent Subalgebras And Hom Structures Of3-lie Algebras

The structure of nilpotent3-Lie algebras is very important in the theory of n-Liealgebras. In this paper, we study non-two-step nilpotent3-Lie algebras whose propersubalgebras are two-step nilpotent. It is proved that if a non-two-step nilpotent3-Lie algebraL whose proper subalgebras are two-step nilpotent, then L is nilpotent. If dim L≠3, thendim L≥6and3≤dim L1≤dim L3. And there exists only one class of six dimensionalsuch3-Lie algebra. The paper also studies the realizations of Hom-3-Lie algebras andinfinite dimensional3-Lie algebras from linear functions and linear mappings.The paper is organized as follows:In section1, we introduce the basic concepts of n-Lie algebras.In section2, we study non-two-step nilpotent3-Lie algebras whose proper subalgebrasare two-step nilpotent, and classifications of3-Lie algebras.In section3, we study constructions of Hom-3-Lie.In section4, we study Hom structures of an infinite dimensional3-Lie algebras.