On flat pseudo-Euclidean nilpotent Lie algebras

JOURNAL OF ALGEBRA

卷:537

Article

Boucetta, Mohamed¹  Lebzioui, Hicham² 

[1] Univ Cadi Ayyad, Fac Sci & Tech, BP 549, Marrakech, Morocco

[2] Univ Sultan Moulay Slimane, Ecole Super Technol Khenifra, BP 170, Khenifra, Morocco

关键词: Nilpotent Lie algebras;    Nilpotent Lie groups;    Flat left-invariant metrics;    Double extension;   

DOI  :  10.1016/j.jalgebra.2019.07.018

来源: Elsevier

【 摘 要 】

A flat pseudo-Euclidean Lie algebra is a real Lie algebra with a non degenerate symmetric bilinear form and a left symmetric product whose the commutator is the Lie bracket and such that the left multiplications are skew-symmetric. We show that the center of a flat pseudo-Euclidean nilpotent Lie algebra of signature (2, n - 2) must be degenerate and all flat pseudo-Euclidean nilpotent Lie algebras of signature (2, n -2) can be obtained by using the double extension process from flat Lorentzian nilpotent Lie algebras. We show also that the center of a flat pseudo-Euclidean 2-step nilpotent Lie algebra is degenerate and all these Lie algebras are obtained by using a sequence of double extension from an abelian Lie algebra. In particular, we determine all flat pseudo-Euclidean 2-step nilpotent Lie algebras of signature (2, n-2). The paper contains also some examples in low dimension. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free