
Let 𝔽 be a field of characteristic ≠ 2 and ℒ a finite-dimensional Lie superalgebra over 𝔽. In this article, we study the derivation superalgebra Der(ℒ), the quasiderivation superalgebra QDer(ℒ), and the generalized derivation superalgebra GDer(ℒ) of ℒ, which form a tower Der(ℒ) ⊆ QDer(ℒ) ⊆ GDer(ℒ) ⊆ pl(ℒ), where pl(ℒ) denotes the general linear Lie superalgebra. More precisely, we characterize completely those Lie superalgebras ℒ for which QDer(ℒ) = pl(ℒ). We prove that the quasiderivations of ℒ can be embedded as derivations in a larger Lie superalgebra and, furthermore, when the annihilator of ℒ is equal to zero, we obtain a semidirect sum decomposition of .
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