Irreducible Representations of Quantum Affine Algebras

We construct finite-dimensional representations of the quantum affine algebra associated to the simple finite-dimensional Lie algebra sl(n+1). The module structure is defined on the vector space tensor product of the fundamental representations of the quantum affine algebra. To do this, we find a particular basis of every fundamental representation, consisting of eigenvectors for some of the Drinfeld generators of the algebra. The tensor product of such basis vectors are also eigenvectors, and this simplifies the study of the modules.

We consider the trigonometric solutions of the quantum Yang-Baxter equation with spectral parameters associated to the irreducible finite-dimensional representations of the quantum affine algebra associated to sl(2), using some earlier results.

The explicit comultiplication of the Drinfeld generators is found in the sl(2)-case by solving a functional equation induced by the defining relations in the quantum affine algebra.