Quantum Toroidal Algebra Associated with $\mathfrak {gl}_{m|n}$

Abstract

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.