
Journal of Lie Theory 19 (2009), No. 3, 439462 Copyright Heldermann Verlag 2009 Metacurvature of Riemannian PoissonLie Groups Amine Bahayou Université Kasdi Merbah, B.P. 511  Route de Ghardaia, 30000 Ouargla, Algeria amine.bahayou@gmail.com Mohamed Boucetta Faculté des Sciences et Techniques, BP 549, Guéliz  Marrakech, Morocco mboucetta2@yahoo.fr We study the triple (G, π, <.,.> ) where G is a connected and simply connected Lie group, π and <.,.> are, respectively, a multiplicative Poisson tensor and a left invariant Riemannian metric on G such that the necessary conditions, introduced by Hawkins, to the existence of a non commutative deformation (in the direction of π) of the spectral triple associated to <.,.> are satisfied. We show that the geometric problem of the classification of such triples (G, π, <.,.> ) is equivalent to an algebraic one. We solve this algebraic problem in low dimensions and we give a list of all triples (G, π, <.,.> ) satisfying Hawkins's conditions, up to dimension four. Keywords: PoissonLie groups, contravariant connections, metacurvature, spectral triple. MSC: 58B34; 46I65, 53D17 [ Fulltextpdf (230 KB)] for subscribers only. 