From principal series to finite-dimensional solutions of the Yang-Baxter equation

We start from known solutions of the Yang-Baxter equation with a spectralparameter defined on the tensor product of two infinite-dimensional principalseries representations of the group $\mathrm{SL}(2,\mathbb{C})$ or Faddeev'smodular double. Then we describe its restriction to an irreduciblefinite-dimensional representation in one or both spaces. In this way we obtainvery simple explicit formulas embracing rational and trigonometricfinite-dimensional solutions of the Yang-Baxter equation. Finally, we constructthese finite-dimensional solutions by means of the fusion procedure and find anice agreement between two approaches