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

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Abstract

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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Authors and Affiliations

  1. National Research University Higher School of Economics, Myasnitskaya ul., 20, Moscow, Russian Federation, 101000

    B. Feigin

  2. Landau Institute for Theoretical Physics, pr.Akademika Semenova, 1a, Chernogolovka, Russian Federation, 142432

    B. Feigin

  3. Department of Mathematics, Rikkyo University, Toshima-ku, Tokyo, 171-8501, Japan

    M. Jimbo

  4. Institute for Liberal Arts and Sciences, Kyoto University, Kyoto, 606-8316, Japan

    T. Miwa

  5. Department of Mathematics, Indiana University-Purdue University-Indianapolis, 402 N.Blackford St., LD 270, Indianapolis, IN, 46202, USA

    E. Mukhin

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  1. B. Feigin
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  2. M. Jimbo
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  3. T. Miwa
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  4. E. Mukhin
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Correspondence to E. Mukhin.

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Communicated by H.-T. Yau

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Feigin, B., Jimbo, M., Miwa, T. et al. Finite Type Modules and Bethe Ansatz for Quantum Toroidal \({\mathfrak{gl}_1}\) . Commun. Math. Phys. 356, 285–327 (2017). https://doi.org/10.1007/s00220-017-2984-9

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  • DOI: https://doi.org/10.1007/s00220-017-2984-9

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