Yang-baxterization of the quantum dilogarithm

Abstract

In the present paper, we prove the Yang-Baxter relations for a specific R-matrix dependent on the spectral parameter. For the simpliest Lax operator connected with the trigonometric case of the main commutation relation, this R-matrix plays the role of the fundamental R-matrix in the sense of [1]. Now we describe this Lax operator.Let P and Q be the Heisenberg pair of self-adjoint operators with the commutation relationWe regard the real number 7 as a dimensionless coupling constant and put the Planck constant h equal to 1. We construct the Weyl pair of unitary operatorssatisfying the commutation relationThe quantum space 7/, where P and Q are represented irreducibly, can be realized as L2(IR) with r such that Or = ~r pc(x) _ ~ d i ~ r (4)(coordinate representation). In the sequel, a specific realization of P and Q is not significant. The Lax operator acts in the-tensor product of the quantum space 7/and the auxiliary space 12 = C 2 and can be given by a 2 • 2-matrix whose entries are operators on 7/, U xV)Here, z is a complex number called the spectral parameter. The indices f and a stand for the quantum and auxiliary spaces where Lf, a(x ) acts. In this notation, the main commutation relation (cf.[2]) can conveniently be written in the following form:which is regarded as a relation in 7"l @ 1)1 | ~22. Here, Ral,a2(x) is an operator in the tensor product 1) | 1) = C 4 which can be given by the 4

New solutions of the star–triangle relation with discrete and continuous spin variables

“…This is based on and follows closely Spiridonov's proofs of the elliptic beta integrals [14,35]. 7 One major difference is that rather than considering the integral over a closed contour encircling the origin, the integral is considered over the interval [0, 2π] (the difference between the contours is a simple change of variables). This is done for convenience, primarily to avoid calculations involving roots of complex numbers.…”

The star-triangle relation, lens partition function, and hypergeometric sum/integrals

“…The models considered here are integrable, and satisfy a particular form of the Yang-Baxter equation known as the star-triangle relation. This class of integrable models includes many important examples, such as the two-dimensional Ising [28], Fateev-Zamolodchikov [29], Kashiwara-Miwa [30], Chiral Potts models [31,32], and several others [15,21,24,25,[33][34][35][36][37]. Here a quite general overview of such integrable lattice models and their properties will be given, before moving on to the explicit new examples in Section 3.…”

Quasi-classical expansion of the star-triangle relation and integrable systems on quad-graphs

“…Another solution of the hyperbolic star-triangle equation corresponds to the Boltzmann weights for the Faddeev-Volkov model. In 1995 Faddeev and Volkov obtained [9] a solution of the star triangle relation which, in some sense, could be regarded as an analytic continuation the Fateev-Zamolodchikov solution to negative number of spin states N . Remarkably, the correspoding model of statistical mechanics has positive Boltzmann weights [10], its partition function in the large-lattice limit was calculated in [10,71].…”