Anisotropic Spin Generalization of Elliptic Macdonald–Ruijsenaars Operators and R-Matrix Identities

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter–Belavin R-matrix in the fundamental representation of \(\textrm{GL}_M\). In the scalar case \(M=1\), these operators are the elliptic Macdonald–Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any M is equivalent to a set of R-matrix identities. The proof of identities is based on the properties of elliptic R-matrix including the quantum and the associative Yang–Baxter equations. As an application of our results, we introduce elliptic version of q-deformed Haldane–Shastry model.