A new approach to representations of $3$-Lie algebras and abelian extensions

Abstract

In this paper, we introduce the notion of generalized representation of a $3$-Lie algebra, by which we obtain a generalized semidirect product $3$-Lie algebra. Moreover, we develop the corresponding cohomology theory. Various examples of generalized representations of 3-Lie algebras and computation of 2-cocycles of the new cohomology are provided. Also, we show that a split abelian extension of a 3-Lie algebra is isomorphic to a generalized semidirect product $3$-Lie algebra. Furthermore, we describe general abelian extensions of 3-Lie algebras using Maurer-Cartan elements.

17 pages

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Cited by in corpus (7)

• Symplectic, product and complex structures on 3-Lie algebras
• On non-abelian extensions of 3-Lie algebras
• Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators
• Deformations and extensions of modified $\lambda$-differential $3$-Lie Algebras
• Generalized representations of 3-Hom-Lie algebras
• Cohomologies of 3-Lie algebras with derivations
• Cohomologies and generalized derivation extensions of $n$-Lie algebras