Annals of Combinatorics

, Volume 22, Issue 4, pp 681–710 | Cite as

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

  • Raymond ChengEmail author
  • David M. Jackson
  • Geoff J. Stanley


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.


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

Mathematics Subject Classification

Primary 05E15 Secondary 17B37 16T05 



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.


  1. 1.
    J.W. Alexander, A lemma on a system of knotted curves, Proc. Nat. Acad. Sci. USA. 9 (1923), 93–95.Google Scholar
  2. 2.
    N. Burroughs, The universal \(R\) -matrix for \(U_q{\rm sl}(3)\) and beyond!, Comm. Math. Phys. 127 (1990), no. 1, 109–128.Google Scholar
  3. 3.
    A. L. Cauchy, Œuvres complètes. Series 1. Volume 8, Cambridge Library Collection, Cambridge University Press, Cambridge, 2009, Reprint of the 1893 original.Google Scholar
  4. 4.
    V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge University Press, Cambridge, 1994.Google Scholar
  5. 5.
    V.G. Drinfel’d, Quantum Groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.Google Scholar
  6. 6.
    V.G. Drinfel’d, Almost cocommutative Hopf algebras, Algebra i Analiz 1 (1989), no. 2, 30–46.Google Scholar
  7. 7.
    I.P. Goulden and D. M. Jackson, Combinatorial Enumeration, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983, With a foreword by Gian-Carlo Rota, Wiley-Interscience Series in Discrete Mathematics.Google Scholar
  8. 8.
    J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York-Berlin, 1972, Graduate Texts in Mathematics, Vol. 9.Google Scholar
  9. 9.
    C. Kassel, Quantum Groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.Google Scholar
  10. 10.
    A.N. Kirillov and N. Yu. Reshetikhin, \(q\) -Weyl group and a multiplicative formula for universal \(R\) -matrices, Comm. Math. Phys. 134 (1990), no. 2, 421–431.Google Scholar
  11. 11.
    S.M. Khoroshkin and V. N. Tolstoy, Universal \(R\) -matrix for quantized (super) algebras, Comm. Math. Phys. 141 (1991), no. 3, 599–617.Google Scholar
  12. 12.
    S.Z. Levendorskiĭ and Ya. S. Soĭbel’man, The quantum Weyl group and a multiplicative formula for the \(R\) -matrix of a simple Lie algebra, Funktsional. Anal. i Prilozhen. 25 (1991), no. 2, 73–76.Google Scholar
  13. 13.
    G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, Vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993.Google Scholar
  14. 14.
    T. Ohtsuki, Quantum Invariants: A Study of Knots, 3-Manifolds, and Their Sets, Series on Knots and Everything, Vol. 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002.zbMATHGoogle Scholar
  15. 15.
    M. Rosso, An analogue of P.B.W. theorem and the universal \(R\) -matrix for \(U_h{\rm sl}(N+1)\), Comm. Math. Phys. 124 (1989), no. 2, 307–318.Google Scholar
  16. 16.
    N.Yu. Reshetikhin and V. G. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), no. 1, 1–26.Google Scholar
  17. 17.
    N.Yu. Reshetikhin and V. G. Turaev, Invariants of \(3\) -manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547–597.Google Scholar
  18. 18.
    E.K. Sklyanin, On an algebra generated by quadratic relations, Usp. Mat. Nauk 40 (1985), 214.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Raymond Cheng
    • 1
    Email author
  • David M. Jackson
    • 2
  • Geoff J. Stanley
    • 3
  1. 1.Department of MathematicsColumbia UniversityNew YorkUSA
  2. 2.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  3. 3.Department of PhysicsOxford UniversityOxfordUK

Personalised recommendations