Anisotropic spin generalization of elliptic Ruijsenaars-Macdonald operators and related integrable long-range spin chains

Abstract: 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 Ruijsenaars-Macdonald 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 and prove them for the elliptic Baxter-Belavin R-matrix. We show that the Polychronakos freezing trick can be applied to this model. It provides the commuting set of Hamiltonians for long-range spin chain. We also discuss the trigonometric degenerations based on the XXZ R-matrix. The talk is based on joint work with Andrei Zotov arXiv:2201.05944 arXiv:2202.01177

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

( slides | video )

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BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

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