Finite Type Modules and Bethe Ansatz for Quantum Toroidal gl₁

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl₁ 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 ψ⁺(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 gl₁ 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.