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 YangBaxter 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 Rmatrices 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.

Field analogue of the Ruijsenaars-Schneider model

“…This R-matrix satisfies the quantum Yang-Baxter equation with shifted spectral parameters. Following [51] let us write down its relation to the Baxter-Belavin R-matrix (2.20) in the form of type (B.15): R 12 (z 1 − z 2 ) = g 1 (z 1 + N , q) g 2 (z 2 , q) R ACF 12 ( , z 1 , z 2 | q) g −1 2 (z 2 + N , q) g −1 1 (z 1 , q) . (B.23)…”

“…A similarity of the quantum R-matrix (2.20) with the Kronecker function underlies the so-called associative Yang-Baxter equation. See[51] and references therein.9 Relation (B.27) appears in the 0 order, while in −1 order one has g2(0) = g1(z)O12 g −1 1 (z) g2(0), which is true due to the property (B.21).…”

On factorized Lax pairs for classical many-body integrable systems

“…The proof of the first statement (1.10) is based on the algebraic treatment of g(z, q). Following [76] we mention that the IRF-Vertex correspondence provides the following relation between quantum non-dynamical R-matrix and the intertwining matrix g(z, q):…”