Differential calculus and connections on a quantum plane at a cubic root of unity

We consider the algebra of N x N matrices as a reduced quantum plane on whi ch a finite-dimensional quantum group H acts. This quantum group is a quoti ent of U-q(sl(2, C)), q being an Nth root of unity. Most of the time we sha ll take N = 3; in that case dim(H) = 27. We recall the properties of this a ction and introduce a differential calculus for this algebra: it is a quoti ent of the Wess-Zumino complex. The quantum group H also acts on the corres ponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connecti ons, in the sense of non commutative geometry, that are taken as 1-forms be longing to this differential algebra. By tensoring this differential calcul us with usual forms over space-time, one can construct generalized connecti ons with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.