Quantum super-symmetries (I): Quantum Grassmann super-algebras and a quantum Deligne-Morgan-Manin de Rham complex

We introduce the quantum Manin $\left(m\right|n)$-superspace $A_q^{m|n}$ equipped with a super ⋆-product, and dually, the quantum Grassmann super-algebra ${\Omega }_q\left(m\right|n)$ equipped with the quantum divided power super-structure. The quantum (restricted) Grassmann superalgebra ${\Omega }_q$ and its Manin dual ${\Omega }_q^{!}$ are made into $U_q\left(\mathrm{gl}\right(m\left|n\right))$-module superalgebras, either for q generic, or for q root of unity, via quantum (super) differential operators. We give an explicit realization model for certain simple $U_q\left(\mathrm{gl}\right(m\left|n\right))$-modules and their dimension-formulae, and construct a quantum super de Rham cochain complex of infinite length, which is a quantized version of a classical analogue due to Manin, Deligne-Morgan in the framework of super-symmetry on supermanifolds in gauge field theory.