Finite Type Modules and Bethe Ansatz Equations
Abstract
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 $U_q\mathfrak{g}$.
- Publication:
-
Annales Henri Poincaré
- Pub Date:
- August 2017
- DOI:
- 10.1007/s00023-017-0577-y
- arXiv:
- arXiv:1609.05724
- Bibcode:
- 2017AnHP...18.2543F
- Keywords:
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- Mathematics - Quantum Algebra;
- Mathematical Physics;
- Mathematics - Representation Theory
- E-Print:
- Latex 33 pages