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Register a new userFinite type modules and Bethe ansatz equations
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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 characteristic 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 $U_qmathfrak{g}$.
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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 a simple multiplication operator by taking an appropriate quotient. We expect this approach to be applicable to a wide variety of models.
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 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.
A series of systems of nonlinear equations with affine Weyl group symmetry of type $A^{(1)}_l$ is studied. This series gives a generalization of Painleve equations $P_{IV}$ and $P_{V}$ to higher orders.
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