Finite dimensional quantum group covariant differential calculus on a complex matrix algebra

Using the fact that the algebra M₃(C) of 3x3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Ω(S) on the quantum space S defined by the algebra C^∞(M)⊗M₃(C), where M is a space-time manifold. This calculus is covariant under the action and coaction of finite dimensional dual quantum groups. We study the star structures on these quantum groups and the compatible one in M₃(C). This leads to an invariant scalar product on the later space. We analyse the differential algebra Ω(M₃(C)) in terms of quantum group representations, and consider in particular the space of 1-forms on S since its elements can be considered as generalized gauge fields.