Quasi-triangular Hopf algebras were introduced by Drinfel’d in his construction of solutions to the Yang–Baxter Equation. This algebra is built upon U h ( sl 2 ) , the quantized universal enveloping algebra of the Lie algebra sl 2 . In this paper, combinatorial structure in U h ( sl 2 ) is elicited, and used to assist in highly intricate calculations in this algebra. To this end, a combinatorial methodology is formulated for straightening algebraic expressions to a canonical form in the case n = 1 . We apply this formalism to the quasi-triangular Hopf algebras and obtain a constructive account not only for the derivation of the Drinfel’d’s sR -matrix, but also for the arguably mysterious ribbon elements of U h ( sl 2 ) . Finally, we extend these techniques to the higher-dimensional algebras U h ( sl n + 1 ) . While these explicit algebraic results are well known, our contribution is in our formalism and perspective: our emphasis is on the combinatorial structure of these algebras and how that structure may guide algebraic constructions.