An Example of $Z_N$-Graded Noncommutative Differential Calculus

Abstract

In this work, we consider the algebra $M_N(C)$ of $N\times N$ matrices as a cyclic quantum plane. We also analyze the coaction of the quantum group $\mathcal{F}$ and the action of its dual quantum algebra $\mathcal{H}$ on it. Then, we study the decomposition of $M_N(C)$ in terms of the quantum algebra representations. Finally, we develop the differential algebra of the cyclic group $Z_N$ with $d^N=0$, and treat the particular case N=3.

10 pages