  We study solutions of the Yang-Baxter equation on a tensor product of an
arbitrary finite-dimensional and an arbitrary infinite-dimensional
representations of the rank one symmetry algebra. We consider the cases of the
Lie algebra sl_2, the modular double (trigonometric deformation) and the
Sklyanin algebra (elliptic deformation). The solutions are matrices with
operator entries. The matrix elements are differential operators in the case of
sl_2, finite-difference operators with trigonometric coefficients in the case
of the modular double or finite-difference operators with coefficients
constructed out of Jacobi theta functions in the case of the Sklyanin algebra.
We find a new factorized form of the rational, trigonometric, and elliptic
solutions, which drastically simplifies them. We show that they are products of
several simply organized matrices and obtain for them explicit formulae.
