We introduce and study the quantum toroidal algebra E m|n (q 1 , q 2 , q 3 ) associated with the superalgebra gl m|n with m = 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 E m|n (q 1 , q 2 , q 3 ) → U q gl m|n to the quantum affine algebra associated with the superalgebra gl m|n at level c completed with respect to the homogeneous grading, where q 2 = q 2 and q m−n 3 = c 2 . We also give a bosonic realization of level one E m|n (q 1 , q 2 , q 3 )-modules. Quantum toroidal gl m|nAssume m, n ≥ 1 and m = n. In this section we introduce the quantum toroidal algebra associated with gl m|n , denoted by E m|n , and collect a few properties.2.1. Definition of E m|n . Fix d, q ∈ C × and define

We prove that the quantum toroidal algebras E s associated with different root systems s of gl m|n type are isomorphic. We also show the existence of Miki automorphism of E s , which exchanges the vertical and horizontal subalgebras.To obtain these results, we establish an action of the toroidal braid group on the direct sum ⊕ s E s of all such algebras.

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We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra U q sl(m|n).