Quadratic relations of the deformed $W$-superalgebra ${\cal W}_{q, t}(\mathfrak{sl}(2|1))$

We revisit the free field construction of the deformed $W$-superalgebras ${\cal W}_{q,t}(\mathfrak{sl}(2|1))$ by J. Ding and B. Feigin, {\it Contemp.Math.}{\bf 248}, 83-108 (1998), where the basic $W$-current and screening currents have been found. In this paper we introduce higher $W$-currents and obtain a closed set of quadratic relations among them. These relations are independent of the choice of Dynkin diagrams for the superalgebra $\mathfrak{sl}(2|1)$, though the screening currents are not. This allows us to define ${\cal W}_{q,t}(\mathfrak{sl}(2|1))$ by generators and relations.