Abstract
We introduce and study a category \(\mathcal {O}^\mathrm{fin}_{\mathfrak {b}}\) of modules of the Borel subalgebra \(U_q\mathfrak {b}\) 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 \(\mathcal {O}^\mathrm{fin}_{\mathfrak {b}}\). 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 \(U_q\mathfrak {g}\).
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Communicated by Jean-Michel Maillet.
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Feigin, B., Jimbo, M., Miwa, T. et al. Finite Type Modules and Bethe Ansatz Equations. Ann. Henri Poincaré 18, 2543–2579 (2017). https://doi.org/10.1007/s00023-017-0577-y
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DOI: https://doi.org/10.1007/s00023-017-0577-y