Combinatorial aspects of the quantized universal enveloping algebra of $\mathfrak{sl}_{n+1}(\mathbb{C})$

Quasi-triangular Hopf algebras were introduced by Drinfel'd in his construction of solutions to the Yang--Baxter Equation. This algebra is built upon $\mathcal{U}_h(\mathfrak{sl}_2)$, the quantized universal enveloping algebra of the Lie algebra $\mathfrak{sl}_2$. In this paper, combinatorial structure in $\mathcal{U}_h(\mathfrak{sl}_2)$ 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 $R$-matrix, but also for the arguably mysterious ribbon elements of $\mathcal{U}_h(\mathfrak{sl}_2)$. Finally, we extend these techniques to the higher dimensional algebras $\mathcal{U}_h(\mathfrak{sl}_{n+1})$. 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.