Symplectic, product and complex structures on 3-Lie algebras

Abstract

In this paper, first we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra. Then we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis condition as the integrability condition. A 3-Lie algebra enjoys a product structure if and only if it is the direct sum (as vector spaces) of two subalgebras. We find that there are four types special integrability conditions, and each of them gives rise to a special decomposition of the original 3-Lie algebra. They are also related to $\huaO$-operators, Rota-Baxter operators and matched pairs of 3-Lie algebras. Parallelly, we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special integrability conditions. Finally, we add compatibility conditions between a complex structure and a product structure, between a symplectic structure and a paracomplex structure, between a symplectic structure and a complex structure, to introduce the notions of a complex product structure, a para-Kähler structure and a pseudo-Kähler structure on a 3-Lie algebra. We use 3-pre-Lie algebras to construct these structures. Furthermore, a Levi-Civita product is introduced associated to a pseudo-Riemannian 3-Lie algebra and deeply studied.

37 pages

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Cited by in corpus (9)

• 3-Lie-Rinehart Algebras
• Twisted Rota-Baxter operators on 3-Lie algebras and NS-3-Lie algebras
• Twilled 3-Lie algebras, generalized matched pairs of 3-Lie algebras and O-operators
• Deformations and cohomologies of embedding tensors on 3-Lie algebras
• Deformations and extensions of modified $\lambda$-differential $3$-Lie Algebras
• Hom 3-Lie-Rinehart Algebras
• Quadratic and symplectic structures on 3-(Hom)-$\rho$-Lie algebras
• The constructions of 3-Hom-Lie bialgebras
• Product and complex structures on 3-Bihom-Lie algebras