Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

Abstract

In this paper we propose versions of the associative Yang-Baxter equation and higher order $R$-matrix identities which can be applied to quantum dynamical $R$-matrices. As is known quantum non-dynamical $R$-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for different applications in integrable systems. The dynamical $R$-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is described by the IRF-Vertex transformation. An alternative approach to quantum (semi-)dynamical $R$-matrices and related quantum algebras was suggested by Arutyunov, Chekhov and Frolov (ACF) in their study of the quantum Ruijsenaars-Schneider model. The purpose of this paper is twofold. First, we prove that the ACF elliptic $R$-matrix satisfies the associative Yang-Baxter equation with shifted spectral parameters. Second, we directly prove a simple relation of the IRF-Vertex type between the Baxter-Belavin and the ACF elliptic $R$-matrices predicted previously by Avan and Rollet. It provides the higher order $R$-matrix identities and an explanation of the obtained equations through those for non-dynamical $R$-matrices. As a by-product we also get an interpretation of the intertwining transformation as matrix extension of scalar theta function likewise $R$-matrix is interpreted as matrix extension of the Kronecker function. Relations to the Gervais-Neveu-Felder equation and identities for the Felder's elliptic $R$-matrix are also discussed.

17 pages, minor corrections, reference added

References in corpus (8)

• Relativistic Classical Integrable Tops and Quantum R-matrices
• Rational Top and its Classical R-matrix
• Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations
• Classical integrable systems and soliton equations related to eleven-vertex R-matrix
• Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painleve VI
• Yang-Baxter equations with two Planck constants
• Parameter-dependent associative Yang-Baxter equations and Poisson brackets
• Higher order analogues of unitarity condition for quantum R-matrices

Cited by in corpus (5)

• Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation
• Field analogue of the Ruijsenaars-Schneider model
• On factorized Lax pairs for classical many-body integrable systems
• Non-ultralocal classical r-matrix structure for 1+1 field analogue of elliptic Calogero-Moser model
• Higher rank generalization of 11-vertex rational R-matrix: IRF-Vertex relations and associative Yang-Baxter equation