On R-matrix valued Lax pairs for Calogero-Moser models

Abstract

The article is devoted to the study of $R$-matrix-valued Lax pairs for $N$-body (elliptic) Calogero-Moser models. Their matrix elements are given by quantum ${\mathrm{GL}}_{\tilde{N}}$ $R$-matrices of Baxter-Belavin type. For $\tilde{N}=1$ the widely known Krichever's Lax pair with spectral parameter is reproduced. First, we construct the $R$-matrix-valued Lax pairs for Calogero-Moser models associated with classical root systems. For this purpose we study generalizations of the D'Hoker-Phong Lax pairs. It appeared that in the $R$-matrix-valued case the Lax pairs exist in special cases only. The number of quantum spaces (on which $R$-matrices act) and their dimension depend on the values of coupling constants. Some of the obtained classical Lax pairs admit straightforward extension to the quantum case. In the end we describe a relationship of the $R$-matrix-valued Lax pairs to Hitchin systems defined on ${\mathrm{SL}}_{N\tilde{N}}$ bundles with nontrivial characteristic classes over elliptic curve. We show that the classical analogue of the anisotropic spin exchange operator entering the $R$-matrix-valued Lax equations is reproduced in these models.

28 pages, minor corrections

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Cited by in corpus (9)

• Trigonometric integrable tops from solutions of associative Yang-Baxter equation
• R-matrix-valued Lax pairs and long-range spin chains
• Generalized model of interacting integrable tops
• Anisotropic spin generalization of elliptic Macdonald-Ruijsenaars operators and R-matrix identities
• Elliptic generalization of integrable q-deformed anisotropic Haldane-Shastry long-range spin chain
• Relativistic interacting integrable elliptic tops
• Integrable System of Generalized Relativistic Interacting Tops
• 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