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 $m\ne n$, where the parameters satisfy $q_1{q}_2{q}_3=1$. We give an evaluation map. The evaluation map is a surjective homomorphism of algebras ${\mathcal{E}}_{m|n}(q_1,q_2,q_3)\to {\tilde{U}}_q{\widehat{\mathfrak{gl}}}_{m|n}$ 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.

v1: LaTex, 23 pages. v2: minor corrections