Finite-dimensional modules of quantum affine algebras

We construct finite-dimensional representations of the quantum affine algebra associated to the affine Lie algebra . We define an explicit action of the Drinfeld generators on the vector space tensor product of fundamental representations. The action is defined on a basis of eigenvectors for some of the generators, where the eigenvalues are the coefficients in the Laurent expansion of certain rational functions related to the Drinfeld polynomial corresponding to the module. In these modules we find a maximal submodule and by studying the quotient, we get an explicit description of all finite-dimensional irreducible representations corresponding to Drinfeld polynomials with distinct zeroes. We then describe the associated trigonometric solutions of the quantum Yang–Baxter equation.