ℤ<sub>3</sub>-graded Cartan calculus on 𝒪(ℂ<sub><i>q</i></sub><sup>1|1|1</sup>)

After a brief introduction of the basic concepts of ℤ3-grading, the paper continues by introducing ℤ3-graded differential calculus and ℤ3-graded Cartan calculus. Then, using a matrix R which is a solution of the Yang-Baxter equation, a new ℤ3-graded quantum group and a ℤ3-graded quantum space are introduced. The scheme of the ℤ3-graded differential calculus developed on the algebra of functions on this quantum space is extended to the ℤ3-graded Cartan calculus and all the results obtained are formulated using the matrix R.