On Riemann-Poisson Lie groups

Brahim Alioune; Mohamed Boucetta; Ahmed Sid’Ahmed Lessiad

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 4, page 225-247
  • ISSN: 0044-8753

Abstract

top
A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in [4]. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.

How to cite

top

Alioune, Brahim, Boucetta, Mohamed, and Sid’Ahmed Lessiad, Ahmed. "On Riemann-Poisson Lie groups." Archivum Mathematicum 056.4 (2020): 225-247. <http://eudml.org/doc/297069>.

@article{Alioune2020,
abstract = {A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in [4]. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.},
author = {Alioune, Brahim, Boucetta, Mohamed, Sid’Ahmed Lessiad, Ahmed},
journal = {Archivum Mathematicum},
keywords = {Lie group; Poisson manifolds; Riemannian metric},
language = {eng},
number = {4},
pages = {225-247},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On Riemann-Poisson Lie groups},
url = {http://eudml.org/doc/297069},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Alioune, Brahim
AU - Boucetta, Mohamed
AU - Sid’Ahmed Lessiad, Ahmed
TI - On Riemann-Poisson Lie groups
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 4
SP - 225
EP - 247
AB - A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in [4]. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.
LA - eng
KW - Lie group; Poisson manifolds; Riemannian metric
UR - http://eudml.org/doc/297069
ER -

References

top
  1. Ait Haddou, M., Boucetta, M., Lebzioui, H., Left-invariant Lorentzian flat metrics on Lie groups, J. Lie Theory 22 (1) (2012), 269–289. (2012) MR2933940
  2. Boucetta, M., 10.1016/S0764-4442(01)02132-2, C.R. Acad. Sci. Paris Sér. I 333 (2001), 763–768. (2001) MR1868950DOI10.1016/S0764-4442(01)02132-2
  3. Boucetta, M., 10.1016/S1631-073X(03)00079-7, C.R. Acad. Sci. Paris, Sér. I 336 (2003), 423–428. (2003) MR1979358DOI10.1016/S1631-073X(03)00079-7
  4. Boucetta, M., 10.1016/j.difgeo.2003.10.013, Differential Geom. Appl. 20 (2004), 279–291. (2004) MR2053915DOI10.1016/j.difgeo.2003.10.013
  5. Boucetta, M., On the Riemann-Lie algebras and Riemann-Poisson Lie groups, J. Lie Theory 15 (1) (2005), 183–195. (2005) MR2115235
  6. Deninger, C., Singhof, W., 10.1023/A:1015652906096, Ann. Global Anal. Geom. 21 (2002), 377–399. (2002) MR1910458DOI10.1023/A:1015652906096
  7. Dufour, J.P., Zung, N.T., Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, 2005. (2005) MR2178041
  8. Fernandes, R.L., 10.4310/jdg/1214341648, J. Differential Geom. 54 (2000), 303–366. (2000) MR1818181DOI10.4310/jdg/1214341648
  9. Ha, K.Y., Lee, J.B., 10.1002/mana.200610777, Math. Nachr. 282 (2009), 868–898. (2009) MR2530885DOI10.1002/mana.200610777
  10. Hawkin, E., 10.4310/jdg/1193074900, J. Differential Geom. 77 (2007), 385–424. (2007) MR2362320DOI10.4310/jdg/1193074900
  11. Milnor, J., 10.1016/S0001-8708(76)80002-3, Adv. Math. 21 (1976), 293–329. (1976) MR0425012DOI10.1016/S0001-8708(76)80002-3
  12. Ovando, G., Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory 16 (2006), 371–391. (2006) MR2197598
  13. Vaisman, I., Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, vol. 118, Birkhäuser, Berlin, 1994. (1994) Zbl0810.53019MR1269545

NotesEmbed ?

top

You must be logged in to post comments.