### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 028, 34 pages      arXiv:1411.7595      https://doi.org/10.3842/SIGMA.2016.028

### From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation

Dmitry Chicherin a, Sergey E. Derkachov b and Vyacheslav P. Spiridonov c
a) LAPTH, UMR 5108 du CNRS, associée à l'Université de Savoie, Université de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux, France
b) St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
c) Laboratory of Theoretical Physics, JINR, Dubna, Moscow region, 141980, Russia

Received November 17, 2015, in final form March 04, 2016; Published online March 11, 2016

Abstract
We start from known solutions of the Yang-Baxter equation with a spectral parameter defined on the tensor product of two infinite-dimensional principal series representations of the group $\mathrm{SL}(2,\mathbb{C})$ or Faddeev's modular double. Then we describe its restriction to an irreducible finite-dimensional representation in one or both spaces. In this way we obtain very simple explicit formulas embracing rational and trigonometric finite-dimensional solutions of the Yang-Baxter equation. Finally, we construct these finite-dimensional solutions by means of the fusion procedure and find a nice agreement between two approaches.

Key words: Yang-Baxter equation; principal series; modular double; fusion.

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