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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Supported by the NSF grant DMS-0601912, and by DARPA.
- 2.
Supported by an NSF postdoctoral fellowship.
- 3.
Elements of these spaces are elements of the associative algebra ClN.
References
J. Birman, Braids, links, and mapping class groups, Ann Math Studies, No. 82. Princeton University Press (1974).
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.
F. Constantinescu, F. Toppan, On the linearized Artin braid representation. J. Knot Theory Ramifications 2, no. 4 (1993), 399–412.
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.
N. Geer, B. Patureau-Mirand, V. Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145 no. 1 (2009), 196–212.
L. H. Kauffman, H. Saleur, Free fermions and the Alexander–Conway polynomial, Comm. Math. Phys. 141, no. 2 (1991), 293–327.
P. P. Kulish, Quantum Lie superalgebras and supergroups, Problems of Modern Quantum Field Theory (Alushta, 1989) Springer, 1989, pp. 14–21.
J. Murakami, A state model for multi-variable Alexander polynomial, Pacific J. Math. 157, no. 1 (1993), 109–135.
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.
N. Reshetikhin, V. Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127, no. 1 (1990), 1–26.
M. Rosso, Alexander polynomial and Koszul resolution, Algebra Monpellier Announcements, 1999.
V. Turaev, Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 1994.
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).
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.
N. Geer, B. Patureau-Mirand, V. Turaev, Modified quantum dimensions and re-normalized link invariants, Compos. Math. 145, no. 1 (2009), 196–212.
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Reshetikhin, N., Stroppel, C., Webster, B. (2012). Schur–Weyl-Type Duality for Quantized gl(1|1), the Burau Representation of Braid Groups, and Invariants of Tangled Graphs. In: Itenberg, I., Jöricke, B., Passare, M. (eds) Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol 296. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8277-4_16
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8277-4_16
Published:
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-8276-7
Online ISBN: 978-0-8176-8277-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)