Quantum toroidal algebra associated with \(\mathfrak{gl}_{m|n}\) (Q2040641)

The quantum toroidal algebra associated with \(\mathfrak{gl}_m\) has been found to have many applications in various branches of mathematics and mathematical physics. In the article under review, the authors introduce the quantum toroidal algebra \(\mathcal{E}_{m|n}(q_1,q_2,q_3)\) associated with \(\mathfrak{gl}_{m|n}\) with \(m \neq n\) and \(q_1q_2q_3 = 1\), which generalizes the quantum toroidal algebra \(\mathcal{E}_{m|0}(q_1,q_2,q_3)\) associated with \(\mathfrak{gl}_m\). The authors explain that they expect that the algebra \(\mathcal{E}_{m|n}(q_1,q_2,q_3)\) have many properties similar to \(\mathcal{E}_{m|0}(q_1,q_2,q_3)\). In particular, they construct the evaluation map; a surjective algebra homomorphism from \(\mathcal{E}_{m|n}(q_1,q_2,q_3)\) to the quantum affine algebra associated with \(\mathfrak{gl}_{m|n}\).