Generalized Derivations Of 3-Lie Algebras

We main study the structure of 3-Lie algebras.Generalized derivations, quasiderivations and quasicentroid of 3-algebras are introduced, and basic relations between them are studied. Structures of quasiderivations and quasicentroid of 3-Lie algebras, which contains a maximal diagonalized tours, are systematically investigated. The main results are: for all 3-Lie algebra A, 1) the generalized derivation algebra GDer(A) is the sum of quasiderivation algebra QDer(A) and quasicentroid QΓ(A); 2) quasiderivations of A can be embedded as derivations in a larger algebra; 3) quasiderivation algebra QDer(A)normalizer quasicentroid, that is, [QDer(A), QΓ(A)] ？QΓ(A); 4) if A contains a maximal diagonalized tours T, then QDer(A) and QΓ(A) are diagonalized T-modules, that is, as T-modules,(T, T) semi-simplely acts on QDer(A) and QΓ(A), respectively.