Geometry of the reduced quantum plane

We consider the space M of NxN matrices as a reduced quantum plane and discuss its geometry under the action and coaction of finite dimensional quantum groups (a quotient of U_q(SL(2)), q being an N-th root of unity, and its dual). We also introduce a differential calculus for M: a quotient of the Wess Zumino complex. We shall restrict ourselves to the case N odd and often choose the particular value N=3. The present paper (to appear in the proceedings of the conference "Quantum Groups and Fundamental Physical Applications", Palerme, December 1997) is essentially a short version of math-ph/9807012.