Abstarct: Aabstract Quantum algebra have been receiving much attention in physics and mathematics in the past four decades. In this dissertation we investigated the formalism of the Dunkl derivative operator. The Dunkl operator that we introduced, in addition to considering the normal derivative, also includes parity. Which can be written in two states, even parity and odd parity. In fact, the Dunkl operator is a modified form of the ordinary derivative and is general form than its corresponding initial theory. So that, in the limited case of the Dunkl deformation parameter, the initially form of a normal derivative is recover. One of the most important properties of Dunkl derivatives is that, when we apply the Dunkl derivative in the Schrödinger equation, we have a square root term that is not in the ordinary case for a special n the square root term, removes the degeneracy. It means that in the ordinary case. For a special quantum number n, the state of the system is only odd or even. But in presence of the square root term, we have both odd and even case parities as separate cases. Also, the construction of algebra, which includes the operators of creation and annihilation that are proportional to the degrees of freedom, will lead to the generalization of field theory in this field. The general outline of this proposal is as follows: First, we introduce Dunkl operators, and in the next step, we study the trigonometric and Hamiltonian functions of the harmonic oscillator and paddermitic properties in the presence of Dunkl derivatives. The common functions we consider are a mixture of even and odd states. In this method, in addition to separating specific functions, a new form of it is presented, for example, deformed logger functions, deformed Henkel, and deformed hermitage. Then we calculate the non-relativistic quantum mechanics in the momentum and space to study the energy spectrum of this system for the coordinated oscillator and then the relativistic quantum mechanics in ordinary space in the absence of Dunkl and obtain its energy spectrum and wave function leading a new creation and annihilation operator is introduced that includes the Dunkl deformation parameter. In the next step, we will examine how to construct a curved space in the normal state and then in the presence of the Dunkl operator. Then we study some thermodynamic properties in the presence of Dunkl operator. The algebra related to these operators has been investigated, and baxsed on the introduced algebra, the functions related to the Hilbert space will be introduced in accordance with these functions. At each stage, we compare the results in the limit state (Dunkl parameter=0) with the results of our initial theories. And at the end, an overview of the work ahead is mentioned. The general outline of this proposal is as follows: First, we introduce Dunkl operators, and in the next step, we study the trigonometric and Hamiltonian functions of the harmonic oscillator and paddermitic properties in the presence of Dunkl derivatives. The common functions we consider are a mixture of even and odd states. In this method, in addition to separating specific functions, a new form of it is presented, for example, deformed logger functions, deformed Henkel, and deformed hermitage. Then we calculate the non-relativistic quantum mechanics in the momentum and space to study the energy spectrum of this system for the coordinated oscillator and then the relativistic quantum mechanics in ordinary space in the absence of Dunkl and obtain its energy spectrum and wave function leading a new creation and annihilation operator is introduced that includes the Dunkl deformation parameter. In the next step, we will examine how to construct a curved space in the normal state and then in the presence of the Dunkl operator. Then we study some thermodynamic properties in the presence of Dunkl operator. The algebra related to these operators has been investigated, and baxsed on the introduced algebra, the functions related to the Hilbert space will be introduced in accordance with these functions. At each stage, we compare the results in the limit state (Dunkl parameter=0) with the results of our initial theories. And at the end, an overview of the work ahead is mentioned.