Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

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.Comment: Latex 42 page

Quantum toroidal gl1\mathfrak{gl}_1 algebra : plane partitions

In third paper of the series we construct a large family of representations of the quantum toroidal \gl_1 algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application we obtain a Gelfand-Zetlin type basis for a class of irreducible lowest weight \gl_\infty-modules.Comment: Latex, 38 page