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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.
We establish the method of Bethe ansatz for the XXZ type model obtained from the R-matrix associated to quantum toroidal gl(1). We do that by using shuffle realizations of the modules and by showing that the Hamiltonian of the model is obtained from
We introduce and study a category $text{Fin}$ of modules of the Borel subalgebra of a quantum affine algebra $U_qmathfrak{g}$, where the commutative algebra of Drinfeld generators $h_{i,r}$, corresponding to Cartan currents, has finitely many charact
The affine evaluation map is a surjective homomorphism from the quantum toroidal ${mathfrak {gl}}_n$ algebra ${mathcal E}_n(q_1,q_2,q_3)$ to the quantum affine algebra $U_qwidehat{mathfrak {gl}}_n$ at level $kappa$ completed with respect to the homog
We construct an analog of the subalgebra $Ugl(n)otimes Ugl(m)$ of $Ugl(m+n)$ in the setting of quantum toroidal algebras and study the restrictions of various representations to this subalgebra.
We study solutions of the Bethe ansatz equations of the non-homogeneous periodic XXX model associated to super Yangian $mathrm Y(mathfrak{gl}_{m|n})$. To a solution we associate a rational difference operator $mathcal D$ and a superspace of rational