The regular representation of $U_v({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$

Using quantum differential operators, we construct a super representation of $U_v({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$-submodule isomorphic to the regular representation of $U_v({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$. In this way, we obtain a presentation of $U_v({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n})$ by a basis together with explicit multiplication formulas of the basis elements by generators.