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LOCAL REPRESENTATIONS OF THE LOOP BRAID GROUP

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  10 June 2016

ZOLTÁN KÁDÁR ,
PAUL MARTIN ,
ERIC ROWELL  and
ZHENGHAN WANG
ZOLTÁN KÁDÁR
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom, e-mail: [email protected], [email protected]
PAUL MARTIN
Affiliation:
Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom, e-mail: [email protected], [email protected]
ERIC ROWELL
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA, e-mail: [email protected]
ZHENGHAN WANG
Affiliation:
Microsoft Research, Station Q and Department of Mathematics, University of California, Santa Barbara, CA 93106, USA, e-mail: [email protected], [email protected]
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Abstract

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We study representations of the loop braid group LBn from the perspective of extending representations of the braid group $\mathcal{B}$n. We also pursue a generalization of the braid/Hecke/Temperlely–Lieb paradigm – uniform finite dimensional quotient algebras of the loop braid group algebras.

MSC classification

Primary: 20F36: Braid groups; Artin groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 59 , Issue 2 , May 2017 , pp. 359 - 378
Copyright
Copyright © Glasgow Mathematical Journal Trust 2016 

References

REFERENCES

1. Andruskiewitsch, N. and Schneider, H.-J., Pointed Hopf algebras. New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43 (Cambridge University Press, Cambridge, 2002).Google Scholar
2. Audoux, B., Bellingeri, P., Meilhan, J.-B. and Wagner, E., Homotopy classification of ribbon tubes and welded string links, J. Math. Phys. 55 (2014), 061702.Google Scholar
3. Baez, J. C., Wise, D. K., Crans, A. S., Exotic statistics for strings in 4D BF theory, Adv. Theor. Math. Phys. 11 (5) (2007), 707–749.Google Scholar
4. Bardakov, V. G., The structure of a group of conjugating automorphisms, Algebra Logika 42 (5) (2003), 515–541, 636; translation in Algebra Logic 42(5) (2003), 287–303.Google Scholar
5. Bardakov, V. G., Linear representations of the group of conjugating automorphisms and the braid groups of some manifolds, (Russian) Sibirsk. Mat. Zh. 46 (1) (2005), 17–31; translation in Siberian Math. J. 46(1) (2005), 13–23.Google Scholar
6. Extending representations of braid groups to the automorphism groups of free groups, J. Knot Theory Ramifications 14 (8) (2005), 1087–1098.Google Scholar
7. Barkeshli, M., Bonderson, P., Cheng, M., and Wang, Z., Symmetry, Defects, and Gauging of Topological Phases, arXiv preprint arXiv:1410.4540 (2014).Google Scholar
8. Bar-Natan, D. and Dancso, Zs., Finite type invariants of w-Knotted objects: from Alexander to Kashiwara and Vergne, arXiv preprint arXiv:1309.7155.Google Scholar
9. Birman, J., Braid links and mapping class groups, in Annals of Mathematics Studies, vol. 82 (Princeton, 1975).Google Scholar
10. Birman, J. and Wenzl, H., Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1) (1989), 249–273.Google Scholar
11. Brendle, T. E. and Hatcher, A., Configuration spaces of rings and wickets, Commentarii Matematici Helvetici 88 (1), (2013), 131–162.CrossRefGoogle Scholar
12. Crane, L., Yetter, Louis D., Categorical Construction of 4D Topological Quantum Field Theories, in Quantum Topology (Kauffman, L.H. and Baadhio, R.A., Editors) (World Scientific, Singapore, 1993).Google Scholar
13. Dahm, D. M., A generalisation of braid theory, PhD thesis (Princeton University, 1962).Google Scholar
14. Etingof, P., Rowell, E. C. and Witherspoon, S. J., Braid group representations from quantum doubles of finite groups, Pacific J. Math. 234 (1) (2008), 33–41.Google Scholar
15. Fenn, R., Rimányi, R. and Rourke, C., Some remarks on the braid-permutation group, in Topics in Knot Theory (Erzurum, 1992), 57–68, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 399, Kluwer Acad. Publ., Dordrecht, 1993.Google Scholar
16. Galindo, C. and Rowell, E. C., Braid Representations from Unitary Braided Vector Spaces, J. Math. Phys. arXiv:1312.5557.Google Scholar
17. Goldsmith, D. L., The theory of motion groups, Michigan Math. J. 28 (1) (1981), 3–17.Google Scholar
18. Jones, V. F. R., Braid groups, Hecke algebras and type II1 factors, in Geometric Methods in Operator Algebras (Kyoto, 1983), 242–273, Pitman Res. Notes Math. Ser. 123, Longman Sci. Tech., Harlow, 1986.Google Scholar
19. Jones, V. F. R., Hecke algebra representations of braid groups and link polynomials, Ann. of Math. 126 (2) (1987), 335–388.Google Scholar
20. Kamada, S., Braid presentation of virtual knots and welded knots, Osaka J. Math. 44 (2) (2007), 441–458.Google Scholar
21. Kassel, C. and Turaev, V., Braid Groups, in Graduate Texts in Mathematics, vol. 247 (Springer, New York, 2008).Google Scholar
22. Kauffman, L. H. and Lambropoulo, S., Virtual braids and the L-move, J. Knot Theory Its Ramifications, 15 (2006), 773.Google Scholar
23. Lin, X.-S., The motion group of the unlink and its representations, in Topology and physics: Proceedings of the Nankai International Conference in Memory of Xiao-Song Lin, Tianjin, China, 27-31 July 2007 (Lin, Kevin, Wang, Zhenghan, and Zhang, Weiping, Editors) (World Scientific, Singapore, 2008), 411.Google Scholar
24. McCool, J., On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (6) (1986) 1525–1529.CrossRefGoogle Scholar
25. Martin, P., Potts models and related problems in statistical mechanics (World Scientific, Singapore, 1991).CrossRefGoogle Scholar
26. Martin, P., On Schur-Weyl duality, An Hecke algebras and quantum sl(N), Int. J. Mod. Phys. A7(Suppl.1B) (1992), 645–673.Google Scholar
27. Murakami, J., The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (4) (1987), 745–758.Google Scholar
28. Nayak, C., Simon, S.-H., Stern, A., Freedman, M., and Sarma, S. Das, Non-Abelian anyons and topological quantum computation, Rev. Modern Phys. 80 (3) (2008), 1083.CrossRefGoogle Scholar
29. Poole, D. G., The stochastic group, Amer. Math. Monthly 102 (9) (1995), 798–801.CrossRefGoogle Scholar
30. Rowell, E. and Wang, Z., Localization of unitary braid group representations, Comm. Math. Phys. 311 (3) (2012), 595–615.Google Scholar
31. Turaev, V. G., Quantum Invariants of Knots and Manifolds (de Gruyter, 1994).CrossRefGoogle Scholar
32. Vershinin, V. V., On homology of virtual braids and Burau representation, J. Knot Theory Ramifications 10 (5) (2001), 795–812. Knots in Hellas-98, Vol. 3 (Delphi).CrossRefGoogle Scholar
33. Wang, Z., Topological quantum computation, AMS CBMS-RCSM 112 (2010).Google Scholar
34. Wang, Z., Quantum computing: a quantum group approach, Proceedings of the XXIX International Colloquium on Group-Theoretical Methods in Physics Tianjin, China, 20-26 August 2012. (Bai, C., Gazeau, J.P., Ge, M.L., Editors) (World Scientific, Singapore, 2013), 97–108.Google Scholar
35. Walker, K. and Wang, Z., (3+ 1)-TQFTs and topological insulators, Front. Phys. 7 (2) (2012), 150–159.CrossRefGoogle Scholar