Symplectic, product and complex structures on 3-Lie algebras

JOURNAL OF ALGEBRA

卷:508

Article

Sheng, Yunhe¹  Tang, Rong¹ 

[1] Jilin Univ, Dept Math, Changchun 130012, Jilin, Peoples R China

关键词: 3-Lie algebra;    Symplectic structure;    Complex structure;    Product structure;    Complex product structure;    Para-Kahler structure;    Kaliler structure;   

DOI  :  10.1016/j.jalgebra.2018.05.005

来源: Elsevier

【 摘 要 】

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 0-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-Kahler structure and a pseudoKahler 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. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free