Annales Henri Poincaré

, Volume 18, Issue 8, pp 2543–2579 | Cite as

Finite Type Modules and Bethe Ansatz Equations

  • Boris Feigin
  • Michio Jimbo
  • Tetsuji Miwa
  • Eugene Mukhin
Article

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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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • Boris Feigin
    • 1
    • 2
  • Michio Jimbo
    • 3
  • Tetsuji Miwa
    • 4
  • Eugene Mukhin
    • 5
  1. 1.International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of Economics Russian FederationMoscowRussia
  2. 2.Landau Institute for Theoretical PhysicsChernogolovkaRussia
  3. 3.Department of MathematicsRikkyo UniversityToshima-ku, TokyoJapan
  4. 4.Institute for Liberal Arts and SciencesKyoto UniversityKyotoJapan
  5. 5.Department of MathematicsIndiana University–Purdue University-IndianapolisIndianapolisUSA

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