We introduce and study the quantum toroidal algebra $\mathcal{ E}_{ m|n}(q_1,q_2,q_3)$ associated with the superalgebra $\mathfrak{ gl}_{ m|n}$ with $m\neq n$, where the parameters satisfy $q_1q_2q_3=1$. We give an evaluation map. The evaluation map is a surjective homomorphism of algebras $\mathcal{ E}_{ m|n}(q_1,q_2,q_3) \to \widetilde{ U}_q\,\widehat{ \mathfrak{ gl}}_{ m|n}$ to the quantum affine algebra associated with the superalgebra $\mathfrak{ gl}_{ m|n}$ at level $c$ completed with respect to the homogeneous grading, where $q_2=q^2$ and $q_3^{ m-n}=c^2$. We also give a bosonic realization of level one $\mathcal{ E}_{ m|n}(q_1,q_2,q_3)$-modules.
Categories: math.QA math-ph math.MP
