A differential calculus on the addition of Z3-graded quantum super space elements

Quantum(planes) spaces are not really (planes) spaces, they are a generalization of the concept of classical (planes) spaces. More precisely, a quantum space is a deformation of a space that, for particular values of the deformation parameter, coincides with the space. Quantum group terminology was first used by Drinfeld in 1986 and then many authors made different interpretations on the subject. Quantum space terminology was first used by Manin in 1989. Noncommutative geometry, in recent years, started to play an important role in many different areas of mathematics and mathematical physics. The basic structure leading to the noncommutative geometry is a differential calculus on an associative algebra. The thesis consists of five chapters and the original chapters begin with the fourth chapter. In the first chapter of the basic concepts which are used in the sections such as algebra, tensor product, free algebra, graded algebra, Hopf Algebra, Z3-graded algebra, comodule, bimodule, quantum(super) plane and Z2- graded quantum super space are given. In the second chapter, the literature concerning the subject is given. In the third chapter definition of Z3-graded quantum super space, Z3-graded differential calculus, left(right) covariant bimodule and related theorems which are the basis of the thesis are given. In the fourth chapter an operation was defined in the Z3-graded quantum super space. The elements of this space were called 3-dimension quantum super vector. The sum of two quantum super vectors was considered and the necessary conditions were investigated for the addition to be a quantum super vector. It's shown that the obtained structure has a super Hopf algebra structure. Then, the first and second order differentials of the components of the addition vector, which are accepted as coordinate functions, Cartan-Maurer 1-forms and partial derivatives were identified and the commutation relations between them were found. After then, we obtained the necessary relations and we construct a differential calculus of this structure. Finally, the study is discussed and concluded in the last part of the thesis.