Journal of Lie Theory 22 (2012), No. 3, 757--767
Copyright Heldermann Verlag 2012
On Local Structure of Pseudo-Riemannian Poisson Manifolds and Pseudo-Riemannian Lie Algebras
Zhiqi Chen
School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China
chenzhiqi@nankai.edu.cn
Fuhai Zhu
School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China
zhufuhai@nankai.edu.cn
Pseudo-Riemannian Poisson manifolds and pseudo-Riemannian Lie algebras were introduced by M. Boucetta. In this paper, we prove that all pseudo-Riemannian Lie algebras are solvable. Based on our main result and some properties of pseudo-Riemannian Lie algebras, we classify Riemann-Lie algebras of arbitrary dimension and pseudo-Riemannian Lie algebras of dimension at most 3.
Keywords: Levi decomposition, pseudo-Riemannian Poisson manifold, pseudo-Riemannian Lie algebra.
MSC: 53D17, 22E50, 17D25
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