Finite Type Modules and Bethe Ansatz Equations

We introduce and study a category Obfin of modules of the Borel subalgebra U_qb of a quantum affine algebra U_qg, where the commutative algebra of Drinfeld generators h_{i , r}, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional U_qg 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 Obfin. Among them, we find the Baxter Q_i operators and T_i operators satisfying relations of the form T_iQ_i= ∏ _jQ_j+ ∏ _kQ_k. 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 Q_i operators acting in an arbitrary finite-dimensional representation of U_qg.