  We introduce and study a category $\text{Fin}$ of modules of the Borel
subalgebra of a quantum affine algebra $U_q\mathfrak{g}$, 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 $U_q\mathfrak{g}$ 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 $\text{Fin}$. Among them we find the Baxter $Q_i$ operators and
$T_i$ operators satisfying relations of the form $T_iQ_i=\prod_j Q_j+ \prod_k
Q_k$. 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 $U_q\mathfrak{g}$.
