Quantum toroidal \mathfrak{g}{{\mathfrak{l}}_{1}} and Bethe ansatz

We establish the method of Bethe ansatz for the XXZ type model obtained from the R matrix associated to quantum toroidal g{{l}₁}. We do this by using shuffle realizations of the modules and by showing that the Hamiltonian of the model is obtained from a simple multiplication operator by taking an appropriate quotient. We expect this approach to be applicable to a wide variety of models.