Combinatorial Aspects of the Quantized Universal Enveloping Algebra of $\mathfrak {sl}_{n+1}$

Abstract

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