LOCAL REPRESENTATIONS OF THE LOOP BRAID GROUP
Glasgow mathematical journal, Tome 59 (2017) no. 2, pp. 359-378

Voir la notice de l'article provenant de la source Cambridge University Press

We study representations of the loop braid group LB n 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.
KÁDÁR, ZOLTÁN; MARTIN, PAUL; ROWELL, ERIC; WANG, ZHENGHAN. LOCAL REPRESENTATIONS OF THE LOOP BRAID GROUP. Glasgow mathematical journal, Tome 59 (2017) no. 2, pp. 359-378. doi: 10.1017/S0017089516000215
@article{10_1017_S0017089516000215,
     author = {K\'AD\'AR, ZOLT\'AN and MARTIN, PAUL and ROWELL, ERIC and WANG, ZHENGHAN},
     title = {LOCAL {REPRESENTATIONS} {OF} {THE} {LOOP} {BRAID} {GROUP}},
     journal = {Glasgow mathematical journal},
     pages = {359--378},
     year = {2017},
     volume = {59},
     number = {2},
     doi = {10.1017/S0017089516000215},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000215/}
}
TY  - JOUR
AU  - KÁDÁR, ZOLTÁN
AU  - MARTIN, PAUL
AU  - ROWELL, ERIC
AU  - WANG, ZHENGHAN
TI  - LOCAL REPRESENTATIONS OF THE LOOP BRAID GROUP
JO  - Glasgow mathematical journal
PY  - 2017
SP  - 359
EP  - 378
VL  - 59
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000215/
DO  - 10.1017/S0017089516000215
ID  - 10_1017_S0017089516000215
ER  - 
%0 Journal Article
%A KÁDÁR, ZOLTÁN
%A MARTIN, PAUL
%A ROWELL, ERIC
%A WANG, ZHENGHAN
%T LOCAL REPRESENTATIONS OF THE LOOP BRAID GROUP
%J Glasgow mathematical journal
%D 2017
%P 359-378
%V 59
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000215/
%R 10.1017/S0017089516000215
%F 10_1017_S0017089516000215

[1] 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] 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] 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] 4. Bardakov, V. G., The structure of a group of conjugating automorphisms, Algebra Logika 42 (5) (2003), 515–541, 636; translation in Algebra Logic (5) (2003), 287–303. Google Scholar

[5] 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. (1) (2005), 13–23. Google Scholar

[6] 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] 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] 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] 9. Birman, J., Braid links and mapping class groups, in Annals of Mathematics Studies, vol. 82 (Princeton, 1975). Google Scholar

[10] 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] 11. Brendle, T. E. and Hatcher, A., Configuration spaces of rings and wickets, Commentarii Matematici Helvetici 88 (1), (2013), 131–162. Google Scholar | DOI

[12] 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] 13. Dahm, D. M., A generalisation of braid theory, PhD thesis (Princeton University, 1962). Google Scholar

[14] 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] 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] 16. Galindo, C. and Rowell, E. C., Braid Representations from Unitary Braided Vector Spaces, J. Math. Phys. arXiv:1312.5557. Google Scholar

[17] 17. Goldsmith, D. L., The theory of motion groups, Michigan Math. J. 28 (1) (1981), 3–17. Google Scholar

[18] 18. Jones, V. F. R., Braid groups, Hecke algebras and type II factors, in Geometric Methods in Operator Algebras (Kyoto, 1983), 242–273, Pitman Res. Notes Math. Ser. , Longman Sci. Tech., Harlow, 1986. Google Scholar

[19] 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] 20. Kamada, S., Braid presentation of virtual knots and welded knots, Osaka J. Math. 44 (2) (2007), 441–458. Google Scholar

[21] 21. Kassel, C. and Turaev, V., Braid Groups, in Graduate Texts in Mathematics, vol. 247 (Springer, New York, 2008). Google Scholar

[22] 22. Kauffman, L. H. and Lambropoulo, S., Virtual braids and the L-move, J. Knot Theory Its Ramifications, 15 (2006), 773. Google Scholar

[23] 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] 24. Mccool, J., On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (6) (1986) 1525–1529. Google Scholar | DOI

[25] 25. Martin, P., Potts models and related problems in statistical mechanics (World Scientific, Singapore, 1991). Google Scholar | DOI

[26] 26. Martin, P., On Schur-Weyl duality, A Hecke algebras and quantum sl(N), Int. J. Mod. Phys. A7(Suppl.1B) (1992), 645–673. Google Scholar

[27] 27. Murakami, J., The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (4) (1987), 745–758. Google Scholar

[28] 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. Google Scholar | DOI

[29] 29. Poole, D. G., The stochastic group, Amer. Math. Monthly 102 (9) (1995), 798–801. Google Scholar | DOI

[30] 30. Rowell, E. and Wang, Z., Localization of unitary braid group representations, Comm. Math. Phys. 311 (3) (2012), 595–615. Google Scholar

[31] 31. Turaev, V. G., Quantum Invariants of Knots and Manifolds (de Gruyter, 1994). Google Scholar | DOI

[32] 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). Google Scholar | DOI

[33] 33. Wang, Z., Topological quantum computation, AMS CBMS-RCSM 112 (2010). Google Scholar

[34] 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] 35. Walker, K. and Wang, Z., (3+ 1)-TQFTs and topological insulators, Front. Phys. 7 (2) (2012), 150–159. Google Scholar | DOI

Cité par Sources :