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Journal of Lie Theory 19 (2009), No. 3, 439--462
Copyright Heldermann Verlag 2009

Metacurvature of Riemannian Poisson-Lie Groups

Amine Bahayou
Université Kasdi Merbah, B.P. 511 -- Route de Ghardaia, 30000 Ouargla, Algeria

Mohamed Boucetta
Faculté des Sciences et Techniques, BP 549, Guéliz - Marrakech, Morocco

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: Poisson-Lie groups, contravariant connections, metacurvature, spectral triple.

MSC: 58B34; 46I65, 53D17

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