Communications in Mathematical Physics

, Volume 356, Issue 1, pp 285–327 | Cite as

Finite Type Modules and Bethe Ansatz for Quantum Toroidal \({\mathfrak{gl}_1}\)

  • B. Feigin
  • M. Jimbo
  • T. Miwa
  • E. Mukhin


We study highest weight representations of the Borel subalgebra of the quantum toroidal \({\mathfrak{gl}_1}\) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of ‘finite type’ modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current \({\psi^+(z)}\) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T V,W (u; p) analogous to those of the six vertex model. In our setting T V,W (u; p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal \({\mathfrak{gl}_1}\) with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules V the corresponding transfer matrices, Q(u; p) and \({\mathcal{T}(u;p)}\), are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u; p). Then we show that the eigenvalues of T V,W (u; p) are given by an appropriate substitution of eigenvalues of Q(u; p) into the q-character of V.


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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussian Federation
  2. 2.Landau Institute for Theoretical PhysicsChernogolovkaRussian Federation
  3. 3.Department of MathematicsRikkyo UniversityToshima-kuJapan
  4. 4.Institute for Liberal Arts and SciencesKyoto UniversityKyotoJapan
  5. 5.Department of MathematicsIndiana University-Purdue University-IndianapolisIndianapolisUSA

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