Finite Type Modules and Bethe Ansatz Equations

Abstract

We introduce and study a category OfinbObfin of modules of the Borel subalgebra UqbUqb of a quantum affine algebra UqgUqg, where the commutative algebra of Drinfeld generators hi,rhi,r, corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional UqgUqg 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 OfinbObfin. Among them, we find the Baxter QiQi operators and TiTi operators satisfying relations of the form TiQi=∏jQj+∏kQkTiQi=∏jQj+∏kQk. 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 QiQi operators acting in an arbitrary finite-dimensional representation of UqgUqg.