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The main purpose of this paper is the construction of the R-operator whichacts in the tensor product of two infinite-dimensional representations of theconformal algebra and solves Yang-Baxter equation. We build the R-operator as aproduct of more elementary operators S_1, S_2 and S_3. Operators S_1 and S_3are identified with intertwining operators of two irreducible representationsof the conformal algebra and the operator S_2 is obtained from the intertwiningoperators S_1 and S_3 by a certain duality transformation. There arestar-triangle relations for the basic building blocks S_1, S_2 and S_3 whichproduce all other relations for the general R-operators. In the case of theconformal algebra of n-dimensional Euclidean space we construct the R-operatorfor the scalar (spin part is equal to zero) representations and prove that thestar-triangle relation is a well known star-triangle relation for propagatorsof scalar fields. In the special case of the conformal algebra of the4-dimensional Euclidean space, the R-operator is obtained for more generalclass of infinite-dimensional (differential) representations with nontrivialspin parts. As a result, for the case of the 4-dimensional Euclidean space, wegeneralize the scalar star-triangle relation to the most general star-trianglerelation for the propagators of particles with arbitrary spins.

Conformal algebra: R-matrix and star-triangle relation