Conformal algebra: R-matrix and star-triangle relation

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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 S1, S2 and S3. Operators S1 and S3 are identified with intertwining operators of two irreducible representations of the conformal algebra and the operator S2 is obtained from the intertwining operators S1 and S3 by a certain duality transformation. There are star-triangle relations for the basic building blocks S1, S2 and S3 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.

Keywords

Quantum Groups Conformal and W Symmetry Integrable Hierarchies 
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Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of SciencesSt. PetersburgRussia
  2. 2.Bogoliubov Lab. of Theoretical Physics, JINRDubnaRussia
  3. 3.Chebyshev LaboratorySt.-Petersburg State UniversitySaint-PetersburgRussia
  4. 4.ITPMM.V.Lomonosov Moscow State UniversityMoscowRussia

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