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Статья

Finite Type Modules and Bethe Ansatz Equations

Feigin B. L., Jimbo M., Miwa T., Mukhin E.

We introduce and study a category (Formula presented.) of modules of the Borel subalgebra (Formula presented.) of a quantum affine algebra (Formula presented.), where the commutative algebra of Drinfeld generators (Formula presented.), corresponding to Cartan currents, has finitely many characteristic values. This category is a natural extension of the category of finite-dimensional (Formula presented.) 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 (Formula presented.). Among them, we find the Baxter (Formula presented.) operators and (Formula presented.) operators satisfying relations of the form (Formula presented.). 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 (Formula presented.) operators acting in an arbitrary finite-dimensional representation of (Formula presented.).