Faithful linear representations of the braid groups

Vladimir Turaev

Séminaire Bourbaki (1999-2000)

  • Volume: 42, page 389-409
  • ISSN: 0303-1179

How to cite


Turaev, Vladimir. "Faithful linear representations of the braid groups." Séminaire Bourbaki 42 (1999-2000): 389-409. <>.

author = {Turaev, Vladimir},
journal = {Séminaire Bourbaki},
keywords = {braid groups; Burau representation; Krammer representation},
language = {eng},
pages = {389-409},
publisher = {Société Mathématique de France},
title = {Faithful linear representations of the braid groups},
url = {},
volume = {42},
year = {1999-2000},

AU - Turaev, Vladimir
TI - Faithful linear representations of the braid groups
JO - Séminaire Bourbaki
PY - 1999-2000
PB - Société Mathématique de France
VL - 42
SP - 389
EP - 409
LA - eng
KW - braid groups; Burau representation; Krammer representation
UR -
ER -


  1. [Bi1] S. Bigelow - The Burau representation is not faithful for n ≥ 5, Preprint math.GT/9904100. 
  2. [Bi2] S. Bigelow - Braid groups are linear, Preprint math. GR/0005038. MR1815219
  3. [Bir] J.S. Birman - Braids, links, and mapping class groups, Ann. of Math. Stud., vol. 82, Princeton Univ. Press, Princeton, N.J., 1974. Zbl0305.57013MR375281
  4. [BW] J.S. Birman, H. Wenzl - Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc.313 (1989), 249-273. Zbl0684.57004MR992598
  5. [Br] R. Brauer - On algebras which are connected with the semisimple continuous groups, Ann. of Math.38 (1937), 857-872. Zbl0017.39105MR1503378
  6. [Bu] W. Burau - Über Zopfgruppen und gleichsinnig verdrillte Verkettungen, Abh. Math. Semin. Hamburg. Univ.11 (1935), 179-186. Zbl0011.17801
  7. [Ch] R. Charney - Geodesic automation and growth functions for Artin groups of finite type, Math. Ann.301 (1995), 307-324. Zbl0813.20042MR1314589
  8. [DFG] J.L. Dyer, E. Formanek, E.K. Grossman - On the linearity of automorphism groups of free groups, Arch. Math. (Basel) 38 (1982), 404-409. Zbl0476.20025MR666911
  9. [FLP] A. Fathi, F. Laudenbach, V. Poenaru - Travaux de Thurston sur les surfaces, Astérisque66-67 (1991), Soc. Math. France, Paris. Zbl0446.57010MR1134426
  10. [FP] E. Formanek, C. Procesi - The automorphism group of a free group is not linear, J. Algebra149 (1992), 494-499. Zbl0780.20023MR1172442
  11. [Ga] F.A. Garside - The braid group and other groups, Quart. J. Math.Oxford20 (1969), 235-254. Zbl0194.03303MR248801
  12. [Iv1] N.V. Ivanov - Automorphisms of Teichmüller modular groups, Topology and geometry - Rohlin Seminar, Lecture Notes in Math., vol. 1346, Springer, Berlin-New York (1988), 199-270. Zbl0657.57004MR970079
  13. [Iv2] N.V. Ivanov - Subgroups of Teichmüller modular groups, Transl. of Math. Monographs, vol. 115. Amer. Math. Soc., Providence, RI, 1992. Zbl0776.57001MR1195787
  14. [Jo] V.F.R. Jones - Hecke algebra representations of braid groups and link polynomials, Ann. of Math.126 (1987), 335-388. Zbl0631.57005MR908150
  15. [Ka] C. Kassel - L'ordre de Dehornoy sur les tresses, Séminaire Bourbaki, exposé 865 (novembre 1999). This volume. 
  16. [Kr1] D. Krammer - The braid group B4 is linear, Preprint, Basel (1999). MR1804157
  17. [Kr2] D. Krammer - Braid Groups are linear, Preprint, Basel (2000). MR1888796
  18. [La] R.J. Lawrence - Homological representations of the Hecke algebra, Comm. Math. Phys.135 (1990), 141-191. Zbl0716.20022MR1086755
  19. [LP] D.D. Long, M. Paton - The Burau representation is not faithful for n ≥ 6, Topology32 (1993), 439-447. Zbl0810.57004
  20. [Mi] J. Michel - A note on words in braid monoids, J. Algebra215 (1999), 366- 377. Zbl0937.20017MR1684142
  21. [Mo] J.A. Moody - The Burau representation of the braid group Bn is unfaithful for large n, Bull. Amer. Math. Soc. (N.S.)25 (1991), 379-384. Zbl0751.57005MR1098347
  22. [Mu] J. Murakami - The Kauffman polynomial of links and representation theory, Osaka J. Math.24 (1987), 745-758. Zbl0666.57006MR927059
  23. [We] H. Wenzl - Quantum groups and subfactors of type B, C, and D, Comm. Math. Phys.133 (1990), 383-432. Zbl0744.17021MR1090432
  24. [Zi] M.G. Zinno - On Krammer's Representation of the Braid Group, Preprint math.RT/0002136. 

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