This paper extends the grading and the symmetry of the Z-graded superalgebras of differential operators D*(A) and differential forms ∧*(A) on an associative algebra A. We consider the associative algebra A = C[x1,..., xn] as graded by Zn and consider the algebras D*(A) and ∧*(A) within the tensor category of Zn+1-graded modules. Any braiding/symmetry and any categorical quantization are completely described in terms of this grading.
We explore the quantizations of these algebras with corresponding quantized multiplications. We further describe the braided and quantized braided structures of the algebras’ derivations, curvatures, differential forms, inner derivations and exterior derivations.
We apply these results to the associative algebra A = C[x, y, z, t] and investigate and compare the quantizations of the electromagnetic 2-form and the Maxwell’s equations expressed in terms of differential forms in two settings: in the tensor categories of Z4-graded and Z5-graded modules.