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

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## 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.

## Keywords

*q*-combinatorics Straightening Quantized universal enveloping algebra Ribbon Hopf algebra \(sR\)-matrix

## Mathematics Subject Classification

Primary 05E15 Secondary 17B37 16T05## Notes

### Acknowledgements

DMJ would like to thank Pavel Etingof for useful discussions. We wish to thank an anonymous referee for most valuable suggestions, and an assiduous reading of the paper. RC and DMJ were supported by the Natural Sciences and Engineering Research Council of Canada.

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