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Schur–Weyl-Type Duality for Quantized gl(1|1), the Burau Representation of Braid Groups, and Invariants of Tangled Graphs

  • Nicolai Reshetikhin
  • Catharina Stroppel
  • Ben Webster
Chapter
Part of the Progress in Mathematics book series (PM, volume 296)

Abstract

We show that the Schur–Weyl-type duality between g(1|1) and GL n gives a natural representation-theoretic setting for the relationship between reduced and introduced Burau representations.

Keywords

Braid group Burau representation R-matrix Tangled graph Lie superalgebra 

Notes

Acknowledgements

The authors are grateful for the hospitality shown them by the Mathematics Department of Aarhus University, where this work was completed, and for a Niels Bohr grant from the Danish National Research Foundation.

References

  1. 1.
    J. Birman, Braids, links, and mapping class groups, Ann Math Studies, No. 82. Princeton University Press (1974).Google Scholar
  2. 2.
    J. Birman, D. D. Long, J. Moody, Finite-dimensional representations of Artin’s braid group. In: The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions, Contemporary Math. 169, Amer. Math. Soc. (1994), pp. 123–132.Google Scholar
  3. 3.
    F. Constantinescu, F. Toppan, On the linearized Artin braid representation. J. Knot Theory Ramifications 2, no. 4 (1993), 399–412.Google Scholar
  4. 4.
    N. Geer, B. Patureau-Mirand, An invariant supertrace for the category of representations of Lie superalgebras Pacific J. Math, 238, no. 2 (2008), 331–348.Google Scholar
  5. 5.
    N. Geer, B. Patureau-Mirand, V. Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 no. 1 (2009), 196–212.Google Scholar
  6. 6.
    L. H. Kauffman, H. Saleur, Free fermions and the Alexander–Conway polynomial, Comm. Math. Phys. 141, no. 2 (1991), 293–327.Google Scholar
  7. 7.
    P. P. Kulish, Quantum Lie superalgebras and supergroups, Problems of Modern Quantum Field Theory (Alushta, 1989) Springer, 1989, pp. 14–21.Google Scholar
  8. 8.
    J. Murakami, A state model for multi-variable Alexander polynomial, Pacific J. Math. 157, no. 1 (1993), 109–135.Google Scholar
  9. 9.
    N. Reshetikhin, Quantum Supergroups, Proceedings of the NATO advanced research workshop, Quantum Field Theory, Statistical Mechanics, Quantum Groups, and Topology. (Coral Gables, FL, 1991), 264–282.Google Scholar
  10. 10.
    N. Reshetikhin, V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127, no. 1 (1990), 1–26.Google Scholar
  11. 11.
    M. Rosso, Alexander polynomial and Koszul resolution, Algebra Monpellier Announcements, 1999.Google Scholar
  12. 12.
    V. Turaev, Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 1994.Google Scholar
  13. 13.
    O. Viro, Quantum relatives of the Alexander polynomial, Algebra i Analiz 18:3 (2006) 63-157 (in Russian), St. Petersburg Math. J. 18 no. 3 (2007), 391–457 (in English).Google Scholar
  14. 14.
    N. Geer, B. Patureau-Mirand, An invariant supertrace for the category of representations of Lie superalgebras, Pacific J. Math, 238, no. 2 (2008), 331–348.Google Scholar
  15. 15.
    N. Geer, B. Patureau-Mirand, V. Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145, no. 1 (2009), 196–212.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Nicolai Reshetikhin
    • 1
  • Catharina Stroppel
    • 2
  • Ben Webster
    • 3
  1. 1.University of California at Berkeley and KDV Institute for Mathematics, Universiteit van AmsterdamAmsterdamUSA
  2. 2.Mathematik ZentrumUniversität BonnBonnGermany
  3. 3.Northeastern UniversityBostonUSA

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