Generating functions for the coefficients of the Cremmer–Gervais R-matrices

The coefficients of certain operators on Vcircle times operatorV can be constructed using generating functions. Necessary and sufficient conditions are given for some such operators to satisfy the Yang–Baxter equation. As a corollary we obtain a simple, direct proof that the Cremmer–Gervais R-matrices satisfy the Yang–Baxter equation. This approach also clarifies Cremmer and Gervaisʹs original proof via the dynamical Yang–Baxter equation.