  We consider the algebra of N x N matrices as a reduced quantum plane on which
a finite-dimensional quantum group H acts. This quantum group is a quotient of
U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take
N=3; in that case \dim(H) = 27. We recall the properties of this action and
introduce a differential calculus for this algebra: it is a quotient of the
Wess-Zumino complex. The quantum group H also acts on the corresponding
differential algebra and we study its decomposition in terms of the
representation theory of H. We also investigate the properties of connections,
in the sense of non commutative geometry, that are taken as 1-forms belonging
to this differential algebra. By tensoring this differential calculus with
usual forms over space-time, one can construct generalized connections with
covariance properties with respect to the usual Lorentz group and with respect
to a finite-dimensional quantum group.
