We implement the Bethe anstaz method for the elliptic quantum group $E_{tau,eta}(A_2^{(2)})$. The Bethe creation operators are constructed as polynomials of the Lax matrix elements expressed through a recurrence relation. We also give the eigenvalues of the family of commuting transfer matrices defined in the tensor product of fundamental representations.
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
a simple multiplication operator by taking an appropriate quotient. We expect this approach to be applicable to a wide variety of models.
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 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
eristic values. This category is a natural extension of the category of finite-dimensional $U_qmathfrak{g}$ 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 $text{Fin}$. 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 $U_qmathfrak{g}$.
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
functions $W$. We show that the set of complete factorizations of $mathcal D$ is in canonical bijection with the variety of superflags in $W$ and that each generic superflag defines a solution of the Bethe ansatz equation. We also give the analogous statements for the quasi-periodic supersymmetric spin chains.
We give integral equations for the generating function of the cummulants of the work done in a quench for the Kondo model in the thermodynamic limit. Our approach is based on an extension of the thermodynamic Bethe ansatz to non-equilibrium situation
s. This extension is made possible by use of a large $N$ expansion of the overlap between Bethe states. In particular, we make use of the Slavnov determinant formula for such overlaps, passing to a function-space representation of the Slavnov matrix . We leave the analysis of the resulting integral equations to future work.