An Example of $Z_{N}$-Graded Noncommutative Differential Calculus

TL;DR

This paper develops a noncommutative differential calculus on the cyclic group Z_N, analyzing matrix algebras as quantum planes, and explores quantum group actions and representations, with specific focus on the case N=3.

Contribution

It introduces a Z_N-graded differential calculus on matrix algebras, extending noncommutative geometry to cyclic groups and quantum symmetries.

Findings

Decomposition of matrix algebra into quantum algebra representations

Construction of differential algebra with d^N=0

Explicit analysis for N=3 case

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 ${\cal F}$ and the action of its dual quantum algebra ${\cal H}$ 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.