Metacurvature of Riemannian Poisson-Lie groups

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