We construct the integrable families of the generalized elliptic Ruijsenaars models by use of the Yang–Baxter equation and the reflection equation. The higher order conserved operators of these models are explicitly presented.
The generalized elliptic Ruijsenaars models, which are regarded as a difference analog of the Calogero–Sutherland–Moser models associated with the classical root systems are studied. The integrability and the duality using the fusion procedure of operator-valued solutions of the Yang–Baxter equation and the reflection equation are shown. In particular a new integrable difference operator of type-D is proposed. The trigonometric models are also considered in terms of the representation of the affine Hecke algebra.
On the R-matrix identities related to elliptic anisotropic spin Ruijsenaars–Macdonald operators
We propose and prove a set of identities for the elliptic GLMR -matrix (in the fundamental representation). In the scalar case ( M=1 ), these are elliptic function identities derived by Ruijsenaars as necessary and sufficient conditions for his kernel identity underlying the construction of integral solutions of quantum spinless Ruijsenaars–Schneider model. In this respect, our result can be regarded as a first step toward constructing solutions of the quantum eigenvalue problem for the anisotropic spin Ruijsenaars model.
We explore the reflection–transmission quantum Yang–Baxter equations, arising in factorized scattering theory of integrable models with impurities. The physical origin of these equations is clarified and three general families of solutions are described in detail. Explicit representatives of each family are also displayed. These results allow us to establish, for the first time, a direct relationship with the different previous works on the subject and make evident the advantages of the reflection–transmission algebra as a universal approach to integrable systems with impurities.