Symplectic, product and complex structures on 3-Lie algebras

TL;DR

This paper explores advanced geometric structures on 3-Lie algebras, including phase spaces, product, complex, and symplectic structures, establishing their interrelations and construction methods via 3-pre-Lie algebras.

Contribution

It introduces new notions of phase space, product, and complex structures on 3-Lie algebras, and studies their compatibility and construction using 3-pre-Lie algebras.

Findings

A 3-Lie algebra has a phase space iff it is sub-adjacent to a 3-pre-Lie algebra.

Four special integrability conditions lead to specific decompositions of 3-Lie algebras.

Construction of complex, product, and symplectic structures using 3-pre-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-K\"{a}hler structure and a pseudo-K\"{a}hler 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.