  We study degenerations of the Belavin R-matrices via the infinite dimensional
operators defined by Shibukawa-Ueno. We define a two-parameter family of
generalizations of the Shibukawa-Ueno R-operators. These operators have finite
dimensional representations which include Belavin's R-matrices in the elliptic
case, a two-parameter family of twisted affinized Cremmer-Gervais R-matrices in
the trigonometric case, and a two-parameter family of twisted (affinized)
generalized Jordanian R-matrices in the rational case. We find finite
dimensional representations which are compatible with the elliptic to
trigonometric and rational degeneration. We further show that certain members
of the elliptic family of operators have no finite dimensional representations.
These R-operators unify and generalize earlier constructions of Felder and
Pasquier, Ding and Hodges, and the authors, and illuminate the extent to which
the Cremmer-Gervais R-matrices (and their rational forms) are degenerations of
Belavin's R-matrix.
