Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities

Abstract

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\mathrm{GL}}_M$. 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 generalization of q-deformed Haldane-Shastry model.

arXiv comment38 pages, minor corrections

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Cited by in corpus (5)

• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Lax equations for relativistic $\mathrm{GL}(NM,\mathbb{C})$ Gaudin models on elliptic curve
• The deformed Inozemtsev spin chain
• Supersymmetric generalization of q-deformed long-range spin chains of Haldane-Shastry type and trigonometric GL(N|M) solution of associative Yang-Baxter equation
• R-matrix valued Lax pair for elliptic Calogero-Inozemtsev system and associative Yang-Baxter equations of ${\mathrm{BC}}_n$ type