Yangians: their Representations and Characters

The Yangian Y (g) associated to a finite-dimensional complex simple Lie algebra g is a Hopf algebra deformation of the universal enveloping algebra of the Lie algebra g[u] of polynomial maps C → g (with Lie bracket defined pointwise). It is important to understand the finite-dimensional representations of Y (g). For one thing, they are closely related to rational solutions of the so-called quantum Yang–Baxter equation (see [7] and [5], Chapter 12). For another, they have recently been shown to act as ‘quantum symmetry groups’ of certain integrable systems (see [1]). In [8], Drinfel’d gave a classification of the finite-dimensional irreducible representations of Y (g) analogous to the classification of the finite-dimensional irreducible representations of g in terms of highest weights, but this left open the problem of giving an ‘explicit’ realization of these representations. This was! solved in [3] when g = sl2