Finite Type Modules and Bethe Ansatz Equations

We introduce and study a category of modules of the Borel subalgebra of a quantum affine algebra , where the commutative algebra of Drinfeld generators , corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional modules. In particular, we classify the irreducible objects, discuss their properties, and describe the combinatorics of the q-characters. We study transfer matrices corresponding to modules in . Among them, we find the Baxter operators and operators satisfying relations of the form . We show that these operators are polynomials of the spectral parameter after a suitable normalization. This allows us to prove the Bethe ansatz equations for the zeroes of the eigenvalues of the operators acting in an arbitrary finite-dimensional representation of .