We investigate the kernel space of an integral operator M(g) depending on the spin g and describing an elliptic Fourier transformation. The operator M(g) is an intertwiner for the elliptic modular double formed from a pair of Sklyanin algebras with t
he parameters $eta$ and $tau$, Im$ tau>0$, Im$eta>0$. For two-dimensional lattices $g=neta + mtau/2$ and $g=1/2+neta + mtau/2$ with incommensurate $1, 2eta,tau$ and integers $n,m>0$, the operator M(g) has a finite-dimensional kernel that consists of the products of theta functions with two different modular parameters and is invariant under the action of generators of the elliptic modular double.
We construct an infinite-dimensional solution of the Yang-Baxter equation (YBE) of rank 1 which is represented as an integral operator with an elliptic hypergeometric kernel acting in the space of functions of two complex variables. This R-operator i
ntertwines the product of two standard L-operators associated with the Sklyanin algebra, an elliptic deformation of sl(2)-algebra. It is built from three basic operators $mathrm{S}_1, mathrm{S}_2$, and $mathrm{S}_3$ generating the permutation group of four parameters $mathfrak{S}_4$. Validity of the key Coxeter relations (including the star-triangle relation) is based on the elliptic beta integral evaluation formula and the Bailey lemma associated with an elliptic Fourier transformation. The operators $mathrm{S}_j$ are determined uniquely with the help of the elliptic modular double.
We study properties of transfer matrices in the sl(N) spin chain models. The transfer matrices with an infinite dimensional auxiliary space are factorized into the product of N commuting Baxter Q-operators. We consider the transfer matrices with auxi
liary spaces of a special type (including the finite dimensional ones). It is shown that they can be represented as the alternating sum over the transfer matrices with infinite dimensional auxiliary spaces. We show that certain combinations of the Baxter Q-operators can be identified with the Q-functions which appear in the Nested Bethe Ansatz.
We develop an approach for constructing the Baxter Q-operators for generic sl(N) spin chains. The key element of our approach is the possibility to represent a solution of the the Yang Baxter equation in the factorized form. We prove that such a repr
esentation holds for a generic sl(N) invariant R-operator and find the explicit expression for the factorizing operators. Taking trace of monodromy matrices constructed of the factorizing operators one defines a family of commuting (Baxter) operators on the quantum space of the model. We show that a generic transfer matrix factorizes into the product of N Baxter Q-operators and discuss an application of this representation for a derivation of functional relations for transfer matrices.
