On the Structure of Quantum Toroidal Superalgebra \({{\cal E}_{m|n}}\)

Abstract

Recently the quantum toroidal superalgebra \({{\cal E}_{m|n}}\) associated with \({\mathfrak{g}\mathfrak{l}_{m|n}}\) was introduced by L. Bezerra and E. Mukhin, which is not a quantum Kac–Moody algebra. The quantum toroidal superalgebra \({{\cal E}_{m|n}}\) exploits infinite sequences of generators and relations of the form, which are called Drinfeld realization. In this paper, we use only finite set of generators and relations to define an associative algebra \({\cal E}_{m|n}^\prime \) and show that there exists an epimorphism from \({\cal E}_{m|n}^\prime \) to the quantum toroidal superalgebra \({{\cal E}_{m|n}}\). In particular, the structure of \({\cal E}_{m|n}^\prime \) enjoys some properties like Drinfeld–Jimbo realization.