Generalized Derivations of Lie Superalgebras

Abstract

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 .

Key Words:

• Centroids
• Generalized derivations
• Quasiderivations

2000 Mathematics Subject Classification:

• 17B05
• 17B40

ACKNOWLEDGMENTS

The authors are grateful to the referees for their valuable comments and suggestions on the first version of the article and many thanks to Professor Daoji Meng for his interest and support in this work. This work was supported by NNSF of China (10871057, 10701019).

Notes

Communicated by I. Shestakov.