A New \(\mathbb {Z}_3\)-Graded Quantum Space And Its Geometry

In this article, we initially define an R-matrix with rank 9. Through the application of the "quantum group relation", we derive a \(\mathbb {Z}_3\)-graded quantum group, denoted by \(\widetilde{GL}_q(1|1|1)\), representing the group of \(3\times 3\) matrices. By introducing a \(\mathbb {Z}_3\)-graded quantum space, denoted by \(\widetilde{\mathbb {C}}_q^{1|1|1}\), along with its exterior algebra, we formulate two \(\mathbb {Z}_3\)-graded differential calculi which are covariant with respect to the \(\mathbb {Z}_3\)-graded Hopf algebra of functions on the \(\mathbb {Z}_3\)-graded quantum group \(\widetilde{GL}_q(1|1|1)\).