Conformal algebra: R-matrix and star-triangle relation

Abstract

The main purpose of this paper is the construction of the R-operator which acts in the tensor product of two infinite-dimensional representations of the conformal algebra and solves Yang-Baxter equation. We build the R-operator as a product of more elementary operators S_1, S_2 and S_3. Operators S_1 and S_3 are identified with intertwining operators of two irreducible representations of the conformal algebra and the operator S_2 is obtained from the intertwining operators S_1 and S_3 by a certain duality transformation. There are star-triangle relations for the basic building blocks S_1, S_2 and S_3 which produce all other relations for the general R-operators. In the case of the conformal algebra of n-dimensional Euclidean space we construct the R-operator for the scalar (spin part is equal to zero) representations and prove that the star-triangle relation is a well known star-triangle relation for propagators of scalar fields. In the special case of the conformal algebra of the 4-dimensional Euclidean space, the R-operator is obtained for more general class of infinite-dimensional (differential) representations with nontrivial spin parts. As a result, for the case of the 4-dimensional Euclidean space, we generalize the scalar star-triangle relation to the most general star-triangle relation for the propagators of particles with arbitrary spins.