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 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.

Automated summary

In this paper, a commuting set of matrix-valued difference operators is proposed based on the elliptic Baxter-Belavin R-matrix in the fundamental representation of 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. It is shown 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 the results, an elliptic version of the q-deformed Haldane-Shastry model is introduced.