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Generalized derivations 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, $[QDer(A), QΓ(A)]\subseteq QΓ(A)$; 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.