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On Riemann-Poisson Lie groups. (English) Zbl 07285962
Summary: 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 [M. Boucetta, Differ. Geom. Appl. 20, No. 3, 279–291 (2004; Zbl 1061.53058)]. 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.
53A15 Affine differential geometry
53D17 Poisson manifolds; Poisson groupoids and algebroids
22E05 Local Lie groups
Zbl 1061.53058
Full Text: DOI
[1] Ait Haddou, M.; Boucetta, M.; Lebzioui, H., Left-invariant Lorentzian flat metrics on Lie groups, J. Lie Theory 22 (1) (2012), 269-289
[2] Boucetta, M., Compatibilité des structures pseudo-riemanniennes et des structures de Poisson, C.R. Acad. Sci. Paris Sér. I 333 (2001), 763-768
[3] Boucetta, M., Riemann-Poisson manifolds and Kähler-Riemann foliations, C.R. Acad. Sci. Paris, Sér. I 336 (2003), 423-428
[4] Boucetta, M., Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras, Differential Geom. Appl. 20 (2004), 279-291
[5] Boucetta, M., On the Riemann-Lie algebras and Riemann-Poisson Lie groups, J. Lie Theory 15 (1) (2005), 183-195
[6] Deninger, C.; Singhof, W., Real polarizable hodge structures arising from foliation, Ann. Global Anal. Geom. 21 (2002), 377-399
[7] Dufour, J. P.; Zung, N. T., Poisson Structures and Their Normal Forms, Progress in Mathematics, vol. 242, Birkhäuser Verlag, 2005
[8] Fernandes, R. L., Connections in Poisson Geometry 1: Holonomy and invariants, J. Differential Geom. 54 (2000), 303-366
[9] Ha, K. Y.; Lee, J. B., Left invariant metrics and curvatures on simply connected three dimensional Lie groups, Math. Nachr. 282 (2009), 868-898
[10] Hawkin, E., The structure of noncommutative deformations, J. Differential Geom. 77 (2007), 385-424
[11] Milnor, J., Curvatures of left invariant metrics on Lie Groups, Adv. Math. 21 (1976), 293-329
[12] Ovando, G., Invariant pseudo-Kähler metrics in dimension four, J. Lie Theory 16 (2006), 371-391
[13] Vaisman, I., Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics, vol. 118, Birkhäuser, Berlin, 1994
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