A left-symmetric algebraic approach to left invariant flat (pseudo-)metrics on Lie groups

JOURNAL OF GEOMETRY AND PHYSICS

卷:62

Article

Chen, Zhiqi^1,2  Hou, Dongping^2,3  Bai, Chengming^2,3 

[1] Nankai Univ, Sch Math Sci, Tianjin 300071, Peoples R China

[2] Nankai Univ, LPMC, Tianjin 300071, Peoples R China

[3] Nankai Univ, Chern Inst Math, Tianjin 300071, Peoples R China

关键词: Left invariant metric;    Affine structure;    Left-symmetric algebra;   

DOI  :  10.1016/j.geomphys.2012.03.003

来源: Elsevier

【 摘 要 】

Left invariant flat metrics on Lie groups are revisited in terms of left-symmetric algebras which correspond to affine structures. There is a left-symmetric algebraic approach with an explicit formula to the classification theorem given by Milnor. When the positive definiteness of the metric is replaced by nondegeneracy, there are many more examples of left invariant flat pseudo-metrics, which play important roles in several fields in geometry and mathematical physics. We give certain explicit constructions of these structures. Their classification in low dimensions and some interesting examples in higher dimensions are also given. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free