SMALL REPRESENTATIONS OF QUANTUM AFFINE ALGEBRAS

We characterize the finite-dimensional representations of the quantum affine algebra U(q)(sl(n+1)) (where q is-an-element-of C(x) is not a r oot of unity) which are irreducible as representations of U(q)(sl(n+1) ). We call such representations 'small'. In 1986, Jimbo defined a fami ly of homomorphisms ev(a) from U(q)(sl(n+1)) to (an enlargement of) U( q)(sl(n+1)), depending on a parameter a is-an-element-of C(x). A secon d family, ev(a), can be obtained by a small modification of Jimbo's fo rmulas. We show that every small representation of U(q)(sl(n+1)) is ob tained by pulling back an irreducible representation of U(q)(sl(n+1)) by ev(a) or ev(a) for some a is-an-element-of C(x).