The Extensions Of 3-Lie Algebras And Special Derivations

The extension of 3-Lie algebras and two classes of special derivations of 3-Lie algebras are studied. The main contents are as following:1. The structure of one dimensional extension of 3-Lie algebras is studied, which is con-structed by one dimensional extension of known Lie algebras.2. The (?,?,?)-extension algebra of 3-Lie algebras is discussed, which is constructed from two known 3-Lie algebras, the necessary and sufficient conditions of (?,?,?)-extension 3-Lie algebra are provided. Meanwhile, the tensor 3-Lie algebra A(?)A(?)A for an arbitrary 3-Lie algebra A, and the extension of derivations are discussed, it is proved that a pair of derivations (?,?)? Der(M)×Der(H) can be extended to a derivation of A if and only if the pair of 3-Lie homomorphisms (f?, f?) can be extended to a 3-Lie homomorphism, which is from A to the tensor 3-Lie algebra A(?)A(?)A.3. The idempotent derivations and T-derivations on general 3-Lie algebra A are defined, and it is proved that there exists an idempotent derivation on A if and only if A=I(?)K, where I is an abelian ideal and K is a subalgebra of A. The structure of idempotent derivations and T-derivations of 3-Lie algebras are investigated. Meanwhile, the difference between T-derivations and derivations is illustrated by examples. The relationships between T-derivation algebra T Der(A), derivation algebra Der(A) and inner derivation algebra ad(A) are discussed. It is proved that ad(A) is an ideal of T Der(A), and if A is a perfect 3-Lie algebra over a field F (Ch?F), then centralizers of derivation algebra and inner derivation algebra in the T-derivation algebra are both zero.