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• W3122794498 abstract "We study highest weight representations of the Borel subalgebra of the quantum toroidal 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 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 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." @default.
• W3122794498 created "2021-02-01" @default.
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• W3122794498 date "2017-08-22" @default.
• W3122794498 modified "2023-10-13" @default.
• W3122794498 title "Finite Type Modules and Bethe Ansatz for Quantum Toroidal $${mathfrak{gl}_1}$$ gl 1" @default.
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• W3122794498 doi "https://doi.org/10.1007/s00220-017-2984-9" @default.
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