Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On pseudo-Riemannian Lie algebras


Authors: Zhiqi Chen, Mingming Ren and Fuhai Zhu
Journal: Proc. Amer. Math. Soc. 138 (2010), 2677-2685
MSC (2010): Primary 17B60, 17D25
Published electronically: April 2, 2010
MathSciNet review: 2644884
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show that $ {\mathfrak{g}} {\mathfrak{g}}\subset{\mathfrak{g}}$ if $ {\mathfrak{g}}$ is a pseudo-Riemannian Lie algebra with $ C(\mathfrak{g})\not=0$. Then we show that $ {\mathfrak{g}}{\mathfrak{g}}\subset{\mathfrak{g}}$ when $ \dim\mathfrak{g}=4$, which leads to the classification of the pseudo-Riemannian Lie algebras in dimension 4.


References [Enhancements On Off] (What's this?)

  • 1. M. Boucetta, Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras, Differential Geom. Appl. 20 (2004), no. 3, 279–291. MR 2053915, 10.1016/j.difgeo.2003.10.013
  • 2. Mohamed Boucetta, Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 8, 763–768 (French, with English and French summaries). MR 1868950, 10.1016/S0764-4442(01)02132-2
  • 3. Mohamed Boucetta, On the Riemann-Lie algebras and Riemann-Poisson Lie groups, J. Lie Theory 15 (2005), no. 1, 183–195. MR 2115235
  • 4. Dietrich Burde and Christine Steinhoff, Classification of orbit closures of 4-dimensional complex Lie algebras, J. Algebra 214 (1999), no. 2, 729–739. MR 1680532, 10.1006/jabr.1998.7714
  • 5. Z. Chen and F. Zhu, On pseudo-Riemannian Lie algebra: A class of new Lie admissible algebras, arXiv: 0807.0936, 2008.
  • 6. Rui Loja Fernandes, Connections in Poisson geometry. I. Holonomy and invariants, J. Differential Geom. 54 (2000), no. 2, 303–365. MR 1818181
  • 7. Y. Kang, Z. Chen and Y. Gao, The classification of pseudo-Riemannian Lie algebras in dimension 4, Acta Sci. Natur. Univ. Nankaiensis 42 (2009), 20-22.
  • 8. John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293–329. MR 0425012
  • 9. Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. MR 1269545

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 17B60, 17D25

Retrieve articles in all journals with MSC (2010): 17B60, 17D25


Additional Information

Zhiqi Chen
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: chenzhiqi@nankai.edu.cn

Mingming Ren
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: rmingming@gmail.com

Fuhai Zhu
Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
Email: zhufuhai@nankai.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10325-6
Keywords: Pseudo-Riemannian Lie algebra, solvable Lie algebra
Received by editor(s): October 8, 2009
Received by editor(s) in revised form: November 6, 2009, November 16, 2009, and November 25, 2009
Published electronically: April 2, 2010
Additional Notes: The third author is the corresponding author. He was supported in part by NNSF Grant #10971103.
Communicated by: Martin Lorenz
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.