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arXiv:2201.05944 · Annales Henri Poincaré

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

Mariya Georgievna Matushko, Andrei Vladimirovich Zotov

Published May 3, 2023• 12 citations
arXivHTMLPDF
DOI
10.1007/s00023-023-01316-y

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On the RRR-matrix identities related to elliptic anisotropic spin Ruijsenaars–Macdonald operators
We propose and prove a set of identities for the elliptic GLMGL_MGLM​ RRR -matrix (in the fundamental representation). In the scalar case ( M=1M=1M=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.
Abstract We describe integrable elliptic q-deformed anisotropic long-range spin chain. The derivation is based on our recent construction for commuting anisotropic elliptic spin Ruijsenaars–Macdonald operators. We prove that the Polychronakos freezing trick can be applied to these operators, thus providing the commuting set of Hamiltonians for long-range spin chain constructed by means of the elliptic Baxter-Belavin G L M R -matrix. Namely, we show that the freezing trick is reduced to a set of elliptic function identities, which are then proved. These identities can be treated as conditions for equilibrium position in the underlying classical spinless Ruijsenaars–Schneider model. Trigonometric degenerations are studied as well. For example, in M = 2 case our construction provides q-deformation for anisotropic XXZ Haldane–Shastry model. The standard Haldane–Shastry model and its Uglov’s q-deformation based on U q ( g l ˆ M ) XXZ R-matrix are included into consideration by separate verification.
We propose commuting sets of matrix-valued difference operators in terms of trigonometric GL(N|M)-valued R-matrices thus providing quantum supersymmetric (and possibly anisotropic) spin Ruijsenaars-Macdonald operators. Two types of trigonometric supersymmetric R-matrices are used for this purpose. The first is the one related to the affine quantized algebra Uˆq(gl(N|M)). The second is a graded version of the standard Zn-invariant An−1 type R-matrix. We show that being properly normalized the latter graded R-matrix satisfies the associative Yang-Baxter equation. Next, we discuss construction of long-range spin chains using the Polychronakos freezing trick. As a result we obtain a new family of spin chains, which extends the gl(N|M)-invariant Haldane-Shastry spin chain to q-deformed case with possible presence of anisotropy.