Quantum (dual) Grassmann superalgebra as $\mathcal U_q(\mathfrak{gl}(m|n))$-module algebra and beyond

Abstract

We introduce and define the quantum affine $(m|n)$-superspace (or say quantum Manin superspace) $A_q^{m|n}$ and its dual object, the quantum Grassmann superalgebra $Ω_q(m|n)$. 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 $\textrm{Diff}_q(Ω_q)$ defined over $Ω_q(m|n)$, which is a smash product of the quantum differential Hopf algebra $\mathfrak D_q(m|n)$ (isomorphic to the bosonization of the quantum Manin superspace) and the quantum Grassmann superalgebra $Ω_q(m|n)$. 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 $Ω_q$ is made into the $\mathcal U_q(\mathfrak g)$-module (super)algebra structure,$Ω_q=Ω_q(m|n)$ for $q$ generic, or $Ω_q(m|n, \bold 1)$ for $q$ root of unity, and $\mathfrak g=\mathfrak{gl}(m|n)$ 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 $Ω_q^!$ 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}.