Metacurvature of Riemannian Poisson-Lie Groups

We study the triple (G, , , ) where G is a connected and simply connected Lie group, 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 ) of the spectral triple associated to , are satisfied.We show that the geometric problem of the classification of such triple (G, , , ) is equivalent to an algebraic one.We solve this algebraic problem in low dimensions and we give the list of all (G, , , ) satisfying Hawkins's conditions, up to dimension four.