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

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 ${\mathcal{U}}_h({\mathfrak{sl}}_2)$, 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.

22 pages, comments always welcome!