Quantum (dual) Grassmann superalgebra as ${\mathcal{U}}_q(\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n))$-module algebra and beyond

We introduce and define the quantum affine $(m\mathrm{\mid }n)$-superspace (or say quantum Manin superspace) $A_q^{m\mathrm{\mid }n}$ and its dual object, the quantum Grassmann superalgebra ${\mathrm{\Omega }}_q(m\mathrm{\mid }n)$. Correspondingly, a quantum Weyl algebra $\mathcal W_q(2(m|n))$of$(m|n)$-type is introduced as the quantum differential operators (QDO for short) algebra ${\text{Diff}}_q({\mathrm{\Omega }}_q)$ defined over ${\mathrm{\Omega }}_q(m\mathrm{\mid }n)$, which is a smash product of the quantum differential Hopf algebra ${\mathfrak{D}}_q(m\mathrm{\mid }n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra ${\mathrm{\Omega }}_q(m\mathrm{\mid }n)$. An interested point of this approach here is that even though $\mathcal W_q(2(m|n))$ itself is in general no longer a Hopf algebra, so are some interesting sub-quotients existed inside. This point of view gives us one of main expected results, that is, the quantum (restricted) Grassmann superalgebra ${\mathrm{\Omega }}_q$ is made into the ${\mathcal{U}}_q(\mathfrak{g})$-module (super)algebra structure,${\mathrm{\Omega }}_q={\mathrm{\Omega }}_q(m\mathrm{\mid }n)$ for $q$ generic, or ${\mathrm{\Omega }}_q(m\mathrm{\mid }n,\mathbf{1})$ for $q$ root of unity, and $\mathfrak{g}=\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$ or $\mathfrak {sl}(m|n)$, the general or special linear Lie superalgebra. This QDO approach provides us with explicit realization models for some simple $\mathcal U_q(\mathfrak g)$-modules, together with the concrete information on their dimensions. Similar results hold for the quantum dual Grassmann superalgebra ${\mathrm{\Omega }}_q^{!}$ as ${\mathcal{U}}_q(\mathfrak{g})$-module algebra.In the paper some examples of pointed Hopf algebras can arise from the QDOs, whose idea is an expansion of the spirit noted by Manin in \cite{Ma}, & \cite{Ma1}.